Hygro-thermo-mechanical analysis of masonry: experimental characterization and numerical simulations

Universidade do Minho Escola de Engenharia Rafael Ramírez Álvarez de Lara Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations January 2023 Rafael Ramírez Álvarez de Lara Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations Doctoral Thesis Civil Engineering Work concluded under the supervision of: Professor Paulo B. Lourenço Professor Bahman Ghiassi Professor Paloma Pineda Universidade do Minho Escola de Engenharia January 2023 ii DIREITOS DE AUTOR E CONDIÇÕES DE UTILIZAÇÃO DO TRABALHO POR TERCEIROS Este é um trabalho académico que pode ser utilizado por terceiros desde que respeitadas as regras e boas práticas internacionalmente aceites, no que concerne aos direitos de autor e direitos conexos. Assim, o presente trabalho pode ser utilizado nos termos previstos na licença abaixo indicada. Caso o utilizador necessite de permissão para poder fazer um uso do trabalho em condições não previstas no licenciamento indicado, deverá contactar o autor, através do RepositóriUM da Universidade do Minho. Licença concedida aos utilizadores deste trabalho Atribuição-NãoComercial-CompartilhaIgual CC BY-NC-SA https://creativecommons.org/licenses/by-nc-sa/4.0/ iii Acknowledgements This work would not have been possible without the effort and support of many people who have been directly or indirectly involved in the process. I would like to express my gratitude to all of them. First and foremost, I would like to thank my supervisors for their fundamental assistance, their dedication, and their encouragement. To Professor Paulo B. Lourenço, thank you for giving me the chance to be part of the research group, for being a source of inspiration, and for giving me a push when I needed it. To Professor Bahman Ghiassi, thank you for your guidance, for all the knowledge that you shared, all your suggestions and discussions, and for your infinite patience with me. To Professor Paloma Pineda, thank you for introducing me to the world of scientific research, for taking me under your wing, and for your constant help throughout all these years. I would like to extend my gratitude to all the professionals who have contributed to the development of this investigation; to Doctor Meera Ramesh and Doctor–to–be Georgios Karanikoloudis, for providing material support for the experimental works; to our colleagues at the University of Beira Interior, namely Professor João Castro–Gomes and Erick Grünhäuser Soares, who arranged and performed the mercury intrusion porosimetry tests; to the technicians of the Structures Laboratory at the University of Minho, for their assistance with the experimental campaign; to the COMSOL Support personnel, for their guidance with the software. My gratitude to the Portuguese Foundation for Science and Technology (FCT) for their financial support through the grant agreement SFRH/BD/117114/2016. I would like to thank all the colleagues who became friends along the way: Maria Pia, Giorgos, Gianpaolo, Ioana, Maxime, Telma, Chandan, Nicoletta, Fabio, Sandra, Carolina, Abide… and so many others that they would not fit in just this one page. This experience would not have been the same without you. A special mention to my flatmates in Guimarães: Meera, Xinyu, Leslie, Antonio, Alberto, Elesban, and Pilar. You made Cappu’s house feel like home. Thanks to all my teachers and professors who showed me the value of knowledge and learning, and helped me get where I am. Finally, I would like to dedicate this work to my family for their unconditional love and support. I will be eternally grateful to you. iv STATEMENT OF INTEGRITY I hereby declare having conducted this academic work with integrity. I confirm that I have not used plagiarism or any form of undue use of information or falsification of results along the process leading to its elaboration. I further declare that I have fully acknowledged the Code of Ethical Conduct of the University of Minho. Resumo v Análise higro–termo–mecânica de alvenaria: Caracterização experimental e simulações numéricas Resumo: As construções de alvenaria espalham–se por todo o mundo, não só em estruturas históricas como também em edifícios contemporâneos. As fachadas de alvenaria constituem uma parte principal da envolvente do edifício, enquanto as paredes portantes funcionam como parte integrante do sistema estrutural. Além disso, as paredes externas estão expostas a ações ambientais que afetam a resposta estrutural e produzem degradação a longo prazo. Neste contexto, os processos higrotérmicos são de especial interesse, uma vez que podem levar a danos significativos. Portanto, a prevenção e reparação de danos relacionados com a temperatura e a humidade na alvenaria precisam de uma compreensão acurada de seu comportamento higrotérmico. Esta tese investiga o comportamento higro–termo– mecânico da alvenaria submetida a condições ambientais. A metodologia engloba a caracterização experimental, bem como simulações numéricas do transporte de calor e humidade e sua influência no desempenho mecânico da alvenaria. A investigação centra–se na alvenaria de tijolo com dois tipos de argamassa, nomeadamente de cal hidráulica natural e de cimento. Como resultado do programa experimental, um conjunto consistente de propriedades do material foi obtido e posteriormente usado para validação de modelos numéricos. Em relação às simulações numéricas, um modelo de transporte de humidade por difusividade foi usado para reproduzir os resultados experimentais de absorção e secagem. Foi demonstrado que a difusividade precisa de ser ajustada de acordo com o processo (absorção/secagem). Além disso, a interface tijolo–argamassa introduz uma resistência hidráulica para a absorção. Um modelo higrotérmico totalmente acoplado foi utilizado para simular a transferência de calor e massa em uma parede de alvenaria. O modelo higrotérmico foi alargado para incorporar efeitos mecânicos e um modelo higro–termo–mecânico acoplado unidirecionalmente foi utilizado para analisar a distribuição de tensões em elementos de alvenaria afetados por variações de temperatura e humidade. Demostrou-se que as condições higrotérmicas alteram significativamente a distribuição de tensões internas da estrutura. A investigação apresentada avança o conhecimento do comportamento higro– termo–mecânico da alvenaria e contribui para a caracterização de materiais e estruturas. A aplicação de técnicas de modelação comumente utilizadas para a análise estrutural de alvenaria oferece grandes possibilidades para o estudo de problemas de transporte de calor e humidade. Palavras-chave: alvenaria; análise higro–termo–mecânica; caracterização experimental; material multicamadas; modelação numérica. Abstract vi Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations Abstract: Masonry constructions are spread all around the world, not only in historical structures but in new buildings as well. Masonry facades make up a fundamental part of the building envelope, whereas load–bearing walls work as an integral part of the structural system. Additionally, the external walls are usually exposed to environmental actions that affect the structural response and produce degradation in the long–term. In this context, hygrothermal processes are of special interest since they may lead to significant damage. Thus, prevention and repair of heat– and moisture–related damage in masonry requires a thorough understanding of its hygrothermal behaviour. This thesis investigates the hygro– thermo–mechanical behaviour of masonry subjected to environmental conditions. The methodology encompasses experimental characterization as well as numerical simulations of heat and moisture transport and their influence on the mechanical performance of masonry. The research focuses on brick masonry with two types of mortar, namely natural hydraulic lime and cement mortar. The experimental program included tests on constituent materials and multi–layered masonry specimens. As a result, a consistent dataset of material properties was obtained and later used for input and validation of numerical models. Regarding the numerical simulations, a diffusivity moisture transport model was used to reproduce water absorption and drying in single materials and multi–layered cases. The model was calibrated and validated against the experimental results. It was demonstrated that the diffusivity function needs to be adjusted depending on the process (wetting/drying). Moreover, the brick–mortar interface works as a hydraulic resistance for water absorption. A fully–coupled hygrothermal model was employed to simulate heat and mass transfer in a brick masonry wall. The hygrothermal model was extended to incorporate mechanical effects and a unidirectionally coupled hygro–thermo–mechanical model was used to analyse the stress distribution of masonry elements as affected by temperature and moisture loads. It was shown that the imposed hygrothermal conditions significantly change the internal stress distribution of the structure. The presented research advances our understanding of hygro–thermo–mechanical behaviour of masonry and thus contributes to the characterization of masonry materials and structures. The application of modelling strategies commonly used for the structural analysis of masonry offers great possibilities for the study of heat and moisture transport problems. Keywords: experimental characterization; hygro–thermo–mechanical analysis; masonry; multi–layered material; numerical modelling. Table of contents vii Table of contents Acknowledgements ........................................................................................................................ iii Resumo ............................................................................................................................................. v Abstract ........................................................................................................................................... vi Table of contents ........................................................................................................................... vii List of figures ................................................................................................................................. xiii List of tables................................................................................................................................... xxi List of symbols ............................................................................................................................. xxiii Abbreviations .............................................................................................................................. xxiii Latin symbols ............................................................................................................................. xxiv Greek symbols ............................................................................................................................ xxvi Subscripts ................................................................................................................................. xxvii CHAPTER 1 Introduction .................................................................................................................................... 1 1.1 General framework and motivation.......................................................................................... 1 1.2 Objectives .............................................................................................................................. 3 1.3 Outline of the thesis ............................................................................................................... 5 CHAPTER 2 Literature review ............................................................................................................................ 7 2.1 Hygrothermal analysis of structures: An interdisciplinary approach .......................................... 7 2.2 Hygrothermal behaviour of building materials ......................................................................... 9 2.2.1 Thermal behaviour ....................................................................................................... 9 2.2.2 Hygric behaviour ........................................................................................................ 10 2.2.3 Experimental research on the characterization of hygrothermal material properties ...... 14 2.2.4 Experimental research on the hygrothermal behaviour of multi–layered materials ........ 15 2.3 Hygrothermal boundary conditions ....................................................................................... 19 2.4 Hygrothermal models ........................................................................................................... 19 2.4.1 Heat problem ............................................................................................................. 20 2.4.2 Moisture problem ....................................................................................................... 22 Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations xiv Figure 4.1. Results of the immersion test at atmospheric pressure: (a) brick units; (b) cement mortar (CM) and lime mortar (LM) specimens. ......................................................................... 55 Figure 4.2. Static gravimetric tests results and fitting curves for the sorption isotherms of the studied materials at 23 °C: (a) brick (B), and (b) lime mortar (LM), fitted with Künzel’s model (Künzel, 1995); (c) cement mortar (CM), and (d) lime mortar from masonry joints (LMJ), fitted with Mualem’s model (Mualem, 1976). ................................................................ 56 Figure 4.3. Capillary absorption results for brick units in the three studied directions. ...................... 60 Figure 4.4. Capillary absorption results for brick cubes in the three studied directions. .................... 61 Figure 4.5. Capillary absorption results for cement mortar (CM) and lime mortar (LM) prisms. ......... 62 Figure 4.6. Drying test results for brick cubes in the three studied directions: (a) moisture mass loss as a function of time; (b) moisture content as a function of time. ........................................ 63 Figure 4.7. Drying test results (X-direction) for cement mortar (CM) and lime mortar (LM) cubes: (a) moisture mass loss as a function of time; (b) moisture content as a function of time. 64 Figure 4.8. Coefficient of hygric expansion and fitting curves for the studied materials: (a) brick (B); (b) lime mortar (LM) and cement mortar (CM). .............................................................. 66 Figure 4.9. MIP analysis – Cumulative porosity and percentage of porous volume occupied by each pore size: (a) brick (B); (b) lime mortar (LM); (c) cement mortar (CM); (d) lime mortar from masonry joints (LMJ). .................................................................................................... 67 Figure 4.10. MIP analysis – Percentage of pores according to size: (a) brick (B); (b) lime mortar (LM); (c) cement mortar (CM); (d) lime mortar from masonry joints (LMJ). .............................. 68 Figure 4.11. MIP analysis – Differential intrusion or –d𝑉/d(log𝑑), with 𝑉 the intruded volume and 𝑑 the pore diameter, as a function of pore size: (a) brick (B); (b) lime mortar (LM); (c) cement mortar (CM); (d) lime mortar from masonry joints (LMJ). Note that the vertical axis scale is not constant. ................................................................................................................ 69 Figure 4.12. Experimental points from MIP and static gravimetric tests, and (de)sorption isotherms of the studied materials: (a) brick (B), and (b) lime mortar (LM), fitted with Künzel’s model (Künzel, 1995); (c) cement mortar (CM), and (d) lime mortar from masonry joints (LMJ), fitted with Mualem’s model (Mualem, 1976). ................................................................................ 71 Figure 4.13. Water absorption results for masonry specimens M1 (B+LMJ): (a) LMJ–to–B configuration, M1–M; (b) B–to–LMJ configuration, M1–B. Grey curves represent test results, and the black curve is the average. The location of the interface is estimated from the average volume of each material layer. ...................................................................................................... 72 List of figures xv Figure 4.14. Water absorption results for masonry specimens M2 (B+LMJ+B). Grey curves represent test results and black curves are the corresponding average. The location of the interfaces is estimated from the average volume of each material layer. ............................................ 74 Figure 4.15. Water absorption results for masonry specimens M4 (B+LMJ+B+LMJ+B). Grey curves represent test results and black curves are the corresponding average. The location of the interfaces is estimated from the average volume of each material layer. ......................... 75 Figure 4.16. Drying test results for masonry specimens D1 (B+LMJ): (a) moisture mass loss as a function of time; (b) moisture content as a function of time. Grey curves represent test results and black curves are the corresponding average. The location of the interfaces is estimated from the average volume of the exposed material layer. ......................................................... 76 Figure 5.1. Moisture storage curves used for simulations: (a) brick (B), and (b) lime mortar (LM), fitted with Künzel’s model (Künzel, 1995); (c) cement mortar (CM), and (d) lime mortar from masonry joints (LMJ), fitted with Mualem’s model (Mualem, 1976). ............................... 87 Figure 5.2. Capillary absorption results for single materials: (a) B, X or extrusion direction; (b) B, Y or stretcher direction; (c) B, Z or bed direction; (d) CM, isotropic; (e) LM, isotropic. ............ 89 Figure 5.3. Drying results for single materials: (a) B, X or extrusion direction; (b) B, Y or stretcher direction; (c) B, Z or bed direction; (d) CM, isotropic; (e) LM, isotropic. ........................... 90 Figure 5.4. Capillary absorption results for masonry specimens: (a) M1/M or LMJ–B configuration; (b) M1/B or B–LMJ configuration; (c) M2a or B1–LMJ–B2 configuration; (d) M2b or B2– LMJ–B1 configuration; (e) M4 or B–LMJ–B–LMJ–B configuration. ................................. 94 Figure 5.5. Drying results for masonry specimens: (a) D1/B or drying–from–brick configuration; (b) D1/M or drying–from–mortar configuration. ............................................................. 96 Figure 5.6. Moisture transport simulations to validate the hysteresis model. Case study I: wetting followed by drying. Note that two different scales are used for the horizontal axis. ........ 100 Figure 5.7. Moisture transport simulations to validate the hysteresis model. Case study II: drying followed by wetting. Note that two different scales are used for the horizontal axis. ....... 101 Figure 5.8. Modelling strategies for masonry. Adapted from D’Altri et al. (2018), following Lourenço (1996), and Petracca et al. (2017). ............................................................................. 102 Figure 5.9. Capillary absorption simulated using different modelling strategies: (a) continuous micro– modelling, M2; (b) continuous micro–modelling M4; (c) discrete micro–modelling, M2; (d) discrete micro–modelling M4 (e) macro–modelling, M2; (f) macro–modelling, M4. . 105 Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations xvi Figure 5.10. Evolution of internal relative humidity in masonry wallettes simulated using different modelling strategies: (a) detailed micro–modelling; (b) continuous micro–modelling; (c) discrete micro–modelling; (d) macro–modelling...................................................... 109 Figure 6.1. Schematic geometrical configuration of the brick masonry wall: (a) transversal cross– section; (b) vertical cross–section; (c) bond arrangement and brick blocks geometry. Dimensions in mm. .................................................................................................... 120 Figure 6.2. Steady–state analysis (SS). Temperature distribution across the studied three–wythe brick masonry wall: (a) joints with natural hydraulic lime mortar (LMJ); (b) joints with cement mortar (CM). ............................................................................................................... 124 Figure 6.3. Steady–state analysis (SS). Relative humidity distribution across the studied three–wythe brick masonry wall: (a) joints with natural hydraulic lime mortar (LMJ); (b) joints with cement mortar (CM). ............................................................................................................... 125 Figure 6.4. Time–dependent analysis (TD1). Temperature evolution across the studied three–wythe brick masonry wall: (a) joints with natural hydraulic lime mortar (LMJ); (b) joints with cement mortar (CM). ............................................................................................................... 127 Figure 6.5. Time–dependent analysis (TD1). Evolution of the relative humidity across the studied three– wythe brick masonry wall: (a) joints with natural hydraulic lime mortar (LMJ); (b) joints with cement mortar (CM). .................................................................................................. 128 Figure 6.6. Time–dependent analysis with variable external boundary conditions (TD2). Temperature evolution for different points of the studied three–wythe brick masonry wall: (a) joints with natural hydraulic lime mortar (LMJ); (b) joints with cement mortar (CM). ..................... 129 Figure 6.7. Time–dependent analysis with variable external boundary conditions (TD2). Evolution of the relative humidity for different points of the studied three–wythe brick masonry wall: (a) joints with natural hydraulic lime mortar (LMJ); (b) joints with cement mortar (CM). .............. 130 Figure 6.8. Schematic diagram of the one–way coupled hygro–thermo–mechanical model used in this study. ......................................................................................................................... 134 Figure 6.9. Hygro–thermo–mechanical analysis of the three–wythe brick masonry wall with natural hydraulic lime mortar: (a) imposed boundary conditions and displacement history measured at the top of the wall; (b) deformed shape selected for the study (maximum horizontal displacement). ............................................................................................................ 137 Figure 6.10. Hygro–thermo–mechanical analysis of the three–wythe brick masonry wall with cement mortar: (a) imposed boundary conditions and displacement history measured at the top of List of figures xvii the wall; (b) deformed shape selected for the study (maximum horizontal displacement). .................................................................................................................................. 138 Figure 6.11. Hygro–thermo–mechanical analysis of the three–wythe brick masonry wall with natural hydraulic lime mortar. Principal stress distribution obtained for the gravitational loads alone: minimum principal stress, 𝜎3 [MPa], in (a) bricks, and (b) mortar joints; maximum principal stress, 𝜎1 [MPa], in (c) bricks, and (d) mortar joints. .................................................. 139 Figure 6.12. Hygro–thermo–mechanical analysis of the three–wythe brick masonry wall with cement mortar. Principal stress distribution obtained for the gravitational loads alone: minimum principal stress, 𝜎3 [MPa], in (a) bricks, and (b) mortar joints; maximum principal stress, 𝜎1 [MPa], in (c) bricks, and (d) mortar joints. ............................................................. 141 Figure 6.13. Hygro–thermo–mechanical analysis of the three–wythe brick masonry wall with natural hydraulic lime mortar. Principal stress distribution obtained for the hygro–thermo– mechanical case: minimum principal stress, 𝜎3 [MPa], in (a) bricks, and (b) mortar joints; maximum principal stress, 𝜎1 [MPa], in (c) bricks, and (d) mortar joints. ................... 142 Figure 6.14. Hygro–thermo–mechanical analysis of the three–wythe brick masonry wall with cement mortar. Principal stress distribution obtained for the hygro–thermo–mechanical case: minimum principal stress, 𝜎3 [MPa], in (a) bricks, and (b) mortar joints; maximum principal stress, 𝜎1 [MPa], in (c) bricks, and (d) mortar joints. .................................................. 144 Figure 6.15. Hygro–thermo–mechanical analysis of the studied three–wythe brick masonry wall: maximum displacement obtained for the wall with natural hydraulic lime mortar assuming initial conditions equal to (a) external conditions, and (b) internal conditions; maximum displacement for the wall with cement mortar assuming initial conditions equal to (c) external conditions, and (d) internal conditions. ........................................................................ 146 Figure 7.1. Civic Tower of Pavia, Italy: (a) tower and cathedral before the collapse; (b) ruins after the collapse. Images from Binda et al. (2007), Anzani et al. (2009). .................................. 151 Figure 7.2. Civic Tower of Pavia, Italy: (a) geometric survey; (b) outlook of the tower and finite element mesh of the original model by Binda et al. (1992); (c) cross–section of the wall and detail of a retrieved wall fragment showing the external brick cladding. Images adapted from Ferretti & Bazant (2006a), and Binda et al. (2007). ................................................................. 152 Figure 7.3. Outlook of the tower (adapted from Ferretti & Bazant (2006a)) and geometry modelled in the present work. ........................................................................................................ 155 Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations xviii Figure 7.4. Mesh sensitivity study: (a) finite element mesh with maximum mesh size 0.40 m; (b) vertical stress, 𝜎𝑍 [MPa], for self–weight condition in different points of the southwest corner at ground level. ............................................................................................................... 157 Figure 7.5. Temperature records for the city of Pavia: (a) coldest day of the year; (b) warmest day of the year. Adapted from Climate and Average Weather Year–Round in Pavia, Italy (2022). .. 158 Figure 7.6. Vertical stress, 𝜎𝑍 [MPa], at the base of the tower for self–weight condition: (a) results reported by Binda et al. (1992); (b) results obtained with the model developed in the present work. .......................................................................................................................... 160 Figure 7.7. Minimum principal stress, 𝜎3 [MPa], obtained at the base of the tower for different load cases: (a) P.Ext, poorly–ventilated inner space; (b) P.Ext, well–ventilated inner space; (c) P.Int, poorly–ventilated inner space; (d) P.Int, well–ventilated inner space; (e) Core, poorly–ventilated inner space; (f) Core, well–ventilated inner space. ............................ 163 Figure A2.1. Summary of material properties collected from the literature: (a) bulk density; (b) open porosity; (c) specific heat capacity; (d) thermal conductivity; (e) capillary moisture content; (f) capillary absorption coefficient; (g) water vapour resistance (dry cup); (h) water vapour resistance (wet cup); (i) coefficient of thermal expansion; (j) coefficient of hygric expansion. ‘N’ stands for number of data points in each population. Edges of the boxes mark 25th and 75th percentile. ................................................................................................. 182 Figure A3.1. Sensitivity analysis for the capillary moisture content: (a) water absorption; (b) drying. 205 Figure A3.2. Sorption isotherms employed in the sensitivity analysis for the moisture content curve. ................................................................................................................................ 205 Figure A3.3. Sensitivity analysis for the moisture content curve: (a) water absorption; (b) drying. ... 206 Figure A3.4. Sensitivity analysis for the water absorption coefficient: (a) water absorption; (b) drying. 206 Figure A3.5. Sensitivity analysis for the water vapour resistance: (a) water absorption; (b) drying. .. 207 Figure A3.6. Sensitivity analysis for the height of the specimen: (a) water absorption; (b) drying. ... 208 Figure A3.7. Sensitivity analysis for the exposed surface of the specimen: (a) water absorption; (b) drying. ................................................................................................................ 209 Figure A3.8. Sensitivity analysis for the boundary conditions in drying cases: (a) temperature; (b) relative humidity. ................................................................................................. 210 Figure A3.9. Sensitivity analysis for the convective mass transfer coefficient in drying cases. ......... 211 List of figures xix Figure A3.10. Sensitivity analysis for the initial moisture content: (a) water absorption; (b) drying. ... 212 Figure A3.11. Sensitivity analysis for the interface hydraulic resistance in two–layer mono–material assemblies: (a) water absorption; (b) drying. ............................................................. 214 Figure A3.12. Sensitivity analysis for the stacking arrangement of multi–layered mono–material assemblies with imperfect hydraulic contact: (a) fixed first layer, 𝑅𝐼𝐹= 1.0E+09 m/s; (b) fixed first layer, 𝑅𝐼𝐹= 2.0E+09 m/s. ................................................................ 215 Figure A3.13. Sensitivity analysis for the stacking arrangement of multi–layered mono–material assemblies with imperfect hydraulic contact: (a) variable first layer, 𝑅𝐼𝐹= 1.0E+09 m/s; (b) variable first layer, 𝑅𝐼𝐹= 2.0E+09 m/s. ........................................................... 215 Figure A3.14. Sensitivity analysis for the moisture content curve of adjacent materials in a two–layer assembly: (a) water absorption; (b) drying. ................................................................ 217 Figure A3.15. Sensitivity analysis for the water absorption coefficient of adjacent materials in a two–layer assembly: (a) water absorption; (b) drying. ................................................................ 218 Figure A3.16. Sensitivity analysis for the water vapour resistance of adjacent materials in a two–layer assembly: (a) water absorption; (b) drying. ................................................................ 219 Figure A4.1. Hygrothermal simulation of drying using a fixed convective mass transfer coefficient: (a) moisture mass loss as a function of time; (b) temperature at the exposed surface as a function of time. Note: The experimental results of brick cubes drying in the extrusion direction are used as a reference. ............................................................................. 223 Figure A4.2. Hygrothermal simulation of drying considering the Lewis analogy: (a) moisture mass loss as a function of time; (b) temperature at the exposed surface as a function of time. Note: The experimental results of brick cubes drying in the extrusion direction are used as a reference.................................................................................................................. 224 Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations xx This page has been intentionally left blank List of tables xxi List of tables Table 2.1. List of heat and moisture transport mechanisms, causes and driving potentials. Adapted from Künzel (1995). ....................................................................................................... 9 Table 2.2. List of basic characterization experiments and derived material properties. Adapted from Scheffler (2008). ........................................................................................................... 16 Table 2.3. Literature database relating hygrothermal tests in masonry. ........................................... 17 Table 3.1. Number of single–material specimens tested. ............................................................... 39 Table 3.2. Number of masonry specimens tested. ......................................................................... 40 Table 3.3. Environmental conditions for cup tests. ......................................................................... 46 Table 4.1. Results from vacuum saturation tests (CoV between parentheses). ................................ 54 Table 4.2. Fitting parameters and accuracy of the modelled sorption isotherms. ............................. 57 Table 4.3. Results from water absorption tests (CoV between parentheses). ................................... 59 Table 4.4. Coefficient of thermal expansion of the studied materials. .............................................. 65 Table 4.5. Open porosity, ϕo [–], obtained from MIP and vacuum saturation tests. ........................ 72 Table 5.1. Summary of material properties used for moisture transport simulations. ...................... 85 Table 5.2. Updated values of the diffusivity factor γ for drying. ....................................................... 92 Table 5.3. Input parameters used for the different masonry modelling strategies (shaded cells show calibration parameters). .............................................................................................. 104 Table 6.1. Summary of material properties used for the hygrothermal simulations. ....................... 120 Table 6.2. Summary of the initial and boundary conditions used for the hygrothermal simulations. 122 Table 6.3. Summary of mechanical, thermo–mechanical and hygro–mechanical properties used for the simulations. .......................................................................................................... 136 Table 7.1. Mechanical properties determined experimentally from the material remains (Binda et al., 1992). ........................................................................................................................ 153 Table 7.2. Material properties of masonry used in the simulations. ............................................... 156 Table 7.3. Load cases and environmental scenarios studied for the Civic Tower of Pavia. ............. 159 Table A1.1 Thermophysical properties of saturated water. Adapted from Bergman & Lavine (2017). .................................................................................................................................. 177 Table A2.1. Summary of physical and hygrothermal properties of fired–clay brick. .......................... 184 Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations xxii Table A2.2. Summary of mechanical, thermo–mechanical and hygro–mechanical properties of fired– clay brick. ................................................................................................................... 191 Table A2.3. Summary of physical and hygrothermal properties of cement and lime mortars. .......... 193 Table A2.4. Summary of mechanical, thermo–mechanical and hygro–mechanical properties of cement and lime mortars. ....................................................................................................... 200 Table A3.1. Model parameters used for the mono–layered material sensitivity analyses. ................. 204 Table A3.2. Model parameters used for the multi–layered mono–material sensitivity analyses. ....... 214 Table A3.3. Model parameters used for the multi–layered multi–material sensitivity analyses. ........ 216 List of symbols xxiii List of symbols Abbreviations B Fired–clay brick CHE Coefficient of hygric expansion CTE Coefficient of thermal expansion CM Portland cement mortar CoV Coefficient of variation HL Hydrated lime HT Hygrothermal HTM Hygro–thermo–mechanical Int. Interface IRA Initial rate of absorption IUPAC International Union of Pure and Applied Chemistry LB Lower bound LM Natural hydraulic lime mortar LMJ Natural hydraulic lime mortar from masonry bed joints MIP Mercury intrusion porosimetry ND Not defined NHL Natural hydraulic lime NRMSE Normalised root mean square error OPC Ordinary Portland cement REF Reference REV Representative Elementary Volume RH Relative humidity SS Steady–state SW Southwest TD Time–dependent UB Upper bound Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 2 heat and mass transfer mechanisms. Regardless of the chosen approach, a series of material parameters is always necessary to describe the hygrothermal behaviour of a given porous medium. Although the thermal properties of building materials and the corresponding testing methods have been well established in the literature, a reliable characterization of the moisture–related properties still needs more attention (Feng & Janssen, 2016). Numerous works have documented the hygric properties of clay brick, e.g. Kumaran (1996), Roels, Carmeliet, et al. (2003), Scheffler (2008), Aït Ouméziane et al. (2021). However, the existent studies also reveal high intrinsic variability so the extrapolation of literature data to specific real case scenarios is not always straightforward. Moreover, most databases assume isotropic properties and very few references study the possible anisotropic behaviour, which has been nonetheless demonstrated for both moulded and extruded bricks (Gummerson et al., 1980; Krakowiak et al., 2011). Considering the composite nature of masonry, the analysis of mortar and brick– mortar combinations must be included for a consistent definition of the material. The most common mortars used in masonry structures are lime– and cement–based mixes. Lime mortars are more frequently found in historical constructions. However, most characterization studies have focused on cement–based mixes. Similarly, the available studies on brick masonry are mostly concerned with cement–based mortars, and few cases tackle the combination of brick and lime–based mixes, e.g. Groot & Gunneweg (2010a), Nunes et al. (2017), Delgado et al. (2019a), Calle et al. (2019). Moreover, experimental studies generally make use of mortar specimens cast in moulds under controlled laboratory conditions. However, it is known that the different curing conditions obtained between bricks in masonry joints can influence the final quality of the mortar. The study of hygrothermal problems in multi–layered materials has attracted much attention during the last decades. The existing literature on the topic comprises experimental, analytic, and numerical works, which account for the interfacial effects from different points of view. More specifically, the experimental studies on moisture transport in layered composites have been covered to some extent in the literature, e.g. Wilson et al. (1995a, 1995b), Hall & Hoff (2009), Vereecken et al. (2020). However, the experimental works focused on moisture transport in masonry are still limited. In addition, most of the available research has been devoted to liquid water absorption, whereas drying has been scarcely studied. Likewise, only a few investigations have dealt with interface modelling for moisture transport problems in masonry and the conclusions from these works vary from one author to another depending on the adopted methodology and the eventual purpose of the study (Brocken, 1998; Derluyn et al., 2011; Janssen et al., 2012; Vereecken & Roels, 2013; Calle et al., 2019; X. Zhou et al., 2020). Therefore, a unified approach to characterize the brick–mortar interfaces has not been reached yet. Introduction 3 Most of the scientific advances in heat and mass transfer in porous building materials have come from the disciplines of Material Sciences and Building Physics. In this context, the improvement of hygrothermal models and the development of new numerical tools have allowed for more complex and detailed hygrothermal simulations. Nonetheless, the link between the heat and moisture fields and solid mechanics has not been exploited yet. On one hand, experimental studies on structural elements are normally conducted under controlled, standard laboratory conditions. On the other hand, numerical simulations usually neglect the influence of hygrothermal loads and assume ideal conditions. The study of hygro–thermo–mechanical effects has received more attention in concrete structures, e.g. curing of concrete for application in dams (Conceição et al., 2017; Ponce–Farfán et al., 2020). For masonry, some research has been devoted to the mechanical behaviour of walls under extreme scenarios such as high temperatures during fire, see e.g. R. G. Oliveira et al. (2021). However, the influence of temperature and moisture actions caused by normal environmental conditions are usually disregarded and so are the hygrothermal–induced stresses. To date, few studies have approached the mechanical response of masonry as affected by this type of environmental conditions, e.g. Khoshbakht & Lin (2010), Ramézani & Jeong, (2011), Castellazzi, de Miranda, Formica, et al. (2015), and therefore further research is required. 1.2 OBJECTIVES Given the context explained above, the main objective of this thesis is to investigate the hygro–thermo– mechanical behaviour of masonry elements and structures subjected to real environmental conditions. Considering the broad extent of this topic, the investigation is limited to brick masonry walls built with two types of mortar, namely natural hydraulic lime and cement mortar. The focus is set on numerical simulations of heat and moisture transport and their influence on the mechanical performance of masonry. Additionally, the numerical investigation is complemented by the experimental characterization of material properties. The detailed objectives of the work are given below: ▪ Build systematic knowledge for the hygro–thermo–mechanical behaviour of masonry by means of an integrated approach including numerical simulations and experimental tests. The final goal is to understand the hygrothermal response of masonry and how temperature and moisture variations affect the internal stress distribution of masonry structures. ▪ Develop a comprehensive database of hygro–thermo–mechanical material properties. For this investigation, the focus is placed on physical and moisture transport parameters. A commonly Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 4 used set of masonry materials is chosen for experimental characterization, namely solid extruded fired–clay brick, natural hydraulic lime mortar and cement mortar. Among the main goals of the experimental works there is the identification of possible anisotropic response of the materials, different behaviour between the selected mortars, and variation of the mortar properties depending on the curing conditions, namely cast in moulds or cured in masonry joints. ▪ Evaluate the impact of brick–mortar interfaces on the overall hygrothermal behaviour of masonry. The aim is to determine whether the hygrothermal response of masonry is equivalent to the simple addition of its constituent parts or if there are interfacial effects that need to be accounted for as well. If interfacial phenomena are detected, their impact should be calculated through numerical simulations. ▪ Evaluate the existence of hysteretic effects in the moisture storage properties of the material. For this purpose, it is necessary to study both wetting and drying processes and assess whether the materials present different sorption isotherms or moisture retention curves for adsorption and desorption. Similarly, the presence of hysteresis in moisture transport will be assessed by examining the diffusion mechanisms for wetting and drying. ▪ Define the list of input parameters necessary for the chosen hygro–thermo–mechanical model and its main calibration variables. Additionally, a series of sensitivity analyses will be performed to understand how each factor affects the overall response. ▪ Application of different modelling strategies to the study of coupled problems in masonry. To this aim, several techniques commonly used for the structural analysis of masonry will be employed, namely micro– and macro–modelling approaches. ▪ Assessment of the hygrothermal behaviour of masonry by means of numerical simulations. Heat and mass transfer will be analysed separately as well as combined in a coupled hygrothermal scheme. The main purpose is to establish differences and similarities between the heat and moisture fields in the context of masonry materials. In addition, the numerical models will be used to analyse the impact of different modelling choices on the overall response, e.g. the existence of interfacial effects or the selected type of mortar. Assessment of the hygro–thermo–mechanical behaviour of masonry by means of numerical simulations. The hygrothermal model will be extended to incorporate mechanical effects with a one–way or unidirectional coupling scheme. The numerical models will be further employed to evaluate the impact of initial and boundary conditions on the overall response. Among the goals of the numerical studies is the Introduction 5 hygrothermal compatibility analysis of constituent materials according to the induced stress levels resulting from heat and moisture loads. 1.3 OUTLINE OF THE THESIS The contents of the thesis are organized into eight chapters, including the present introduction. Chapter 2 provides a brief review of the existing literature and the current state of knowledge about the topics included in this thesis. The literature review is primarily focused on the most relevant aspects of hygro–thermo–mechanical analysis of porous and multi–layered building materials. First, an overview of heat and mass transfer problems is presented, together with commonly used heat and moisture transport models adopted by various authors. A special emphasis is placed on the different formulations available for the so–called moisture transport diffusivity approaches. Moreover, the basic set of material properties necessary for the simulation models is discussed, and a compilation of hygrothermal properties available in the literature is collected. This is followed by a discussion on hygro–thermo–mechanical models and the coupling possibilities between the different fields. Consequently, the compilation of material parameters is extended to include mechanical, thermo–mechanical and hygro–mechanical properties. Finally, the review is concluded with relevant hygrothermal and hygro–thermo–mechanical studies on masonry materials. Chapter 3 presents the experimental methodology adopted for material characterization. The experimental investigation is mainly focused on the definition of the physical and hygric properties of the selected materials as well as relevant thermo–mechanical and hygro–mechanical parameters. First, a description of the chosen materials (extruded fired–clay brick, natural hydraulic lime mortar, and ordinary Portland cement mortar) is provided. Then, the configuration of the single–material and multi–layered masonry specimens used throughout the study is introduced. The chapter concludes with a detailed description of the experimental procedures. The results obtained from the experimental studies are reported in Chapter 4. At first, the results of the material characterization tests on constituent materials are presented. Special attention is drawn towards the orthotropic nature of extruded fired–clay brick as well as to the differences between the two studied types of mortar. Secondly, the tests performed on multi–layered masonry specimens are discussed. Thus, the characterization of the brick–mortar interface is done on the basis of capillary absorption and drying tests. It is noted that the experimental results presented in this chapter are employed as input and validation data for subsequent simulations. Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 6 Chapter 5 addresses the numerical analysis of moisture transport phenomena in brick masonry. First, a moisture transport diffusivity model is introduced and described in detail. Afterwards, a series of numerical simulations is presented. In particular, capillary absorption and drying processes are studied at the scale of the constituent materials, and then, the analyses are extended to multi–layered cases to evaluate the impact of brick–mortar interfaces. In this context, different model parameters are selected as variables for calibration and the results are validated against experimental data following an iterative fitting procedure. Additionally, a modelling strategy is implemented to capture the hysteresis observed between adsorption and desorption (wetting/drying). Lastly, the proposed model is extended to different modelling approaches commonly used for the structural analysis of masonry. Chapter 6 presents the simulation of different hygrothermal phenomena and their relationship with the mechanical behaviour of masonry components. Initially, the moisture transport model validated in the previous chapter is linked to the thermal field and a fully–coupled hygrothermal model is proposed. Then, hygrothermal simulations using different environmental conditions are performed to study the response of a brick masonry wall. Subsequently, the analyses are extended to incorporate mechanical effects. Thus, a one–way coupled hygro–thermo–mechanical model is presented, and its application is demonstrated on the previously studied brick masonry wall. Chapter 7 builds upon the hygro–thermo–mechanical model presented in the previous chapter and demonstrates its application to simulate the structural behaviour of a full–scale building. For this purpose, a historic masonry tower is selected as a case study. Moreover, different environmental scenarios are evaluated in order to assess the influence of temperature and moisture variations on the mechanical behaviour of the structure. The main conclusions of the developed research are summarized in Chapter 8, together with suggestions and proposals for future works. Finally, a series of Appendices is provided to supply additional information and extend the main concepts introduced throughout the thesis. Literature review 7 CHAPTER 2 Literature review This chapter is devoted to the available literature concerning the theoretical and practical aspects of hygro–thermo–mechanical analyses of porous and multi–layered building materials. An initial overview is provided with focus on the specific features of hygrothermal analyses for civil engineering applications and the hygrothermal properties of porous building materials used in masonry constructions. Then, a summary of relevant mathematical models and case studies is presented, with a succinct revision of the assumptions, main contributions, limitations and drawbacks of each approach. Finally, the review is extended to the hygro–thermo–mechanical models and studies available in the literature. 2.1 HYGROTHERMAL ANALYSIS OF STRUCTURES: AN INTERDISCIPLINARY APPROACH A number of structural components, such as load–bearing masonry walls, make up an essential part of the building envelope system and actively respond to the changes in the environment between indoor and outdoor conditions, namely temperature, air pressure and humidity. This results in a constant exchange of energy and mass (dry air, water vapour, liquid water) through the building component. It is known that temperature and moisture variations in porous materials are related to a series of mechanical effects and degradation mechanisms, such as internal stresses, deformations, volumetric changes, cracks, etc. Moreover, hygrothermal actions in multi–layered components may cause further damage due to the presence of interfaces between dissimilar materials. For instance, deformation mismatches between the constituent components, debonding and cracking at the interface are typical results of multi–layered structures exposed to temperature and humidity fluctuations. Therefore, it is of great importance to identify the main heat and moisture sources that affect the building structures as well as the different transport mechanisms associated with those hygrothermal actions. Moreover, it is necessary to understand how these actions affect the mechanical behaviour of the materials and under which conditions they may cause damage. The main thermal effects on building structures are linked to temperature gradients between indoor and outdoor conditions as well as heat gain by solar radiation. On the other hand, the moisture transport mechanisms through the building envelope depend on the physical state of the water and the moisture source (Figure 2.1). Water can affect a building component in liquid form as rising damp, rain or roof– water leakage. At the same time, moisture may move through the structure as water vapour and condense Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 8 on the external/internal surfaces or at the interfaces between materials in the case of multi–layered components. In addition, the source of moisture may be internal, e.g. related to the formation process of materials such as mortar or concrete. Figure 2.1. Schematic diagrams showing the effect and distribution of moisture in the cross–section of an exposed masonry wall. Adapted from Künzel (1995). Table 2.1 provides a summary of the heat and moisture transport mechanisms that may take place in building components. It must be noted that not all the listed parameters are of interest to the present study. Several transport phenomena and their associated mechanisms will be neglected, namely airflow through the structure, and convection effects based on total pressure differences. The same applies to gravity effects, electrical fields and ion concentration gradients on moisture transport. In addition, freezing is outside the scope of the intended range of temperatures for the current research, and for analogous reasons, high–temperatures (fire conditions) are out of the discussion as well. In the existing literature, it is possible to find numerous hygrothermal (HT) models to calculate the simultaneous heat and moisture transport in building materials and multi–layered components. However, the number of studies devoted to the coupled hygro–thermo–mechanical (HTM) analysis is still scarce. Hygro–thermo–mechanics is the generalisation of a triply coupled field of the single fields of temperature, moisture, and displacement (Szekeres, 2014). Hence, the focus of HTM studies is set on the mechanics of those materials that are thermoscopic, hygroscopic, and deformable. According to Straube & Burnett (2001), any HTM analysis must comply with the following categories of information: 1) Geometrical configuration of the element. 2) Material properties. 3) Boundary conditions, and time–domain in the case of time–dependent analysis. 4) Physics of the coupled HTM problem. 5) Performance thresholds (failure criteria). Literature review 9 Each one of the categories mentioned above is also dependent on the consideration of: a) Dimension, namely one–, two–, or three–dimensional (1–D, 2–D, 3–D). b) Time, namely steady–state, quasi–static, or transient analysis. c) Availability and quality of required information. d) Stochastic nature of each data set. Table 2.1. List of heat and moisture transport mechanisms, causes and driving potentials. Adapted from Künzel (1995). Transport mechanism Cause and driving potential Heat transport Heat conduction Temperature Heat radiation Temperature in 4.th power Airflow Total pressure, density differentials Enthalpy flows through moisture movement Vapour diffusion with phase change and liquid transport flows in the temperature field Vapour transport Gas diffusion Vapour pressure (temperature, total pressure) Molecular transport (Knudsen diffusion or effusion) Vapour pressure Solution diffusion Vapour pressure Liquid transport Capillary conduction Capillary pressure Surface diffusion Relative humidity Seepage flow Gravitation Hydraulic flow Total pressure differentials Electrokinesis Electrical fields Osmosis Ion concentration 2.2 HYGROTHERMAL BEHAVIOUR OF BUILDING MATERIALS The thermal and hygric properties of porous materials are generally defined by non–linear functions dependent on the environmental conditions, namely temperature, relative humidity and air pressure. Moreover, there are mutual dependencies between the temperature and moisture fields, which entails an additional coupled problem. It must be noted that most of the available models do not include time– dependent hygrothermal material properties. This means that ageing, physical deterioration and exposure–related damage are generally not accounted for. 2.2.1 Thermal behaviour Two main properties are used to describe the thermal behaviour of a building material, namely heat capacity and thermal conductivity. The specific heat capacity of a dry material, 𝐶𝑝 [J/(kg·K)], is defined as the energy required to increase the temperature of a unit mass of dry material by 1 K. If the material Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 10 is wet, the specific heat capacity, 𝐶 [J/(kg·K)], can be expressed as: 𝐶=𝐶𝑝+𝐶𝑤·(𝑤/𝜌𝑏𝑢𝑙𝑘) (2.1) where 𝑤 [kg/m3] is the moisture content, 𝜌𝑏𝑢𝑙𝑘 [kg/m3] is the bulk density of the material, and 𝐶𝑤 [J/(kg·K)] is the specific heat capacity of liquid water. The specific heat capacity of water has a weak thermal dependence for the range of temperatures of interest in this study. Therefore, it can be taken as a constant, 𝐶𝑤= 4182 J/(kg·K), which corresponds to the specific heat capacity of water at 20 ℃. For a detailed list of saturated water properties, the reader is referred to Appendix 1 at the end of this document. Additionally, it is useful to define the volumetric heat capacity, 𝜌𝐶 [J/(m3·K)], which is the energy required to increase the temperature of a unit volume of material by 1 K. In wet conditions, the volumetric heat capacity can be calculated as: 𝜌𝐶=𝜌𝑏𝑢𝑙𝑘𝐶𝑝+𝑤·𝐶𝑤 (2.2) Along with the heat capacity, the thermal conductivity of a material characterizes its thermal behaviour. In particular, thermal conductivity refers to the ability of the material to conduct heat. Its definition stems from Fourier’s Law for heat conduction, as the ratio between the heat flow at a point and the thermal gradient at that point in the direction of the flow (see Section 2.4.1 Heat conduction ). 2.2.2 Hygric behaviour A building material is dry if it contains no water or only chemically bonded water. In practice, this state is only possible in non–hygroscopic materials or in hygroscopic materials subjected to drying. Otherwise, hygroscopic materials in contact with moist will air adsorb water molecules from the environment within their pore structure until reaching a state of equilibrium with the ambient humidity. Similarly, capillary– active materials in contact with liquid water will absorb moisture by capillary suction until reaching a certain level of saturation. Hydrophobic materials, on the other hand, do not exhibit capillary suction. Depending on the environmental conditions, moisture inside a building material can appear as vapour, liquid, ice, or a combination of all these phases. The different physical states can seldom be determined separately by direct measurements and phase changes are constantly taking place under natural conditions. Thus, it is only useful to analyse the total sum as a whole or so–called moisture content. Moisture content is the amount of water contained in a material. The moisture content of a building material is always expressed as a ratio, either as mass of moisture per unit volume of the dry material, 𝑤 [kg/m3], as mass of moisture per unit mass of the dry material, 𝑤𝑔 [kg/kg], or as volume Literature review 11 of absorbed moisture per unit volume of the dry material, 𝑤𝑉 [m3/m3]. The moisture content can also be reported as saturation degree or percent of saturation, 𝑆𝑙=𝑤/𝑤𝑠𝑎𝑡 [–], where 𝑤𝑠𝑎𝑡 is the saturation moisture content of the material expressed in any of the formats above. The moisture content of a porous material varies from the dry state to a fully saturated condition, i.e. from null moisture content up to all the open pores filled with water (Figure 2.2). The moisture content corresponding to a fully saturated state is referred to as the saturation or maximum moisture content, 𝑤𝑠𝑎𝑡, and is only achievable if water is forced into the pore structure, e.g. under vacuum. Otherwise, the saturation takes place at a lower moisture content level, defined as the capillary moisture content, 𝑤𝑐𝑎𝑝. Between the dry condition and the saturation state, the moisture content changes to find an equilibrium with the water vapour pressure or relative humidity of the environment. The relation between the ambient humidity and the moisture content within the material is described by the moisture storage function. In general, moisture storage can be reported in the form of sorption isotherms or moisture retention curves. Sorption isotherms describe the moisture content with respect to relative humidity, whereas retention curves define the moisture content with respect to capillary pressure.(1). Numerous analytical expressions are available in the literature for the definition of these moisture storage functions. The different models vary in their flexibility and capacity to represent more complex moisture storage behaviour (e.g. hygroscopic materials and multi–modal curves), and in the number of fitting parameters. A discussion on the different moisture storage models is out of the scope of this review, so the interested reader is referred to specialized works, e.g. Sillers et al. (2001), Carmeliet & Roels (2002). (a) (b) Figure 2.2. Moisture storage functions of hygroscopic and non–hygroscopic porous building materials: (a) sorption isotherm; (b) moisture retention curve. .(1) For wetting liquids in porous materials, 𝑝𝑐≤0, although this sign convention is not always consistent in the literature. In the present work, capillary suction, 𝑝𝑠, is used as the positive–valued capillary pressure, i.e. 𝑝𝑠=−𝑝𝑐. 0.0 0.2 0.4 0.6 0.8 1.0  =  Relative humidity,  [-] Moisture content, w [kg/m3] wcap wsat Moisture range: Hygroscopic Hygroscopic capillary–active material Non–hygroscopic capillary–active material 12345678910 Moisture content, w [kg/m3]  =  log capillary suction, log(ps) [Pa] wsat wcap Overhygroscopic Hygroscopic Capillary Supersaturated wcrit Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 18 Table 2.3 (Continued). Literature database relating hygrothermal tests in masonry Author(s) Specimen type Type of analysis Transfer direction Results Johansson et al. (2014) Masonry wall (brick + cement–lime mortar) Non–isothermal moisture transport 1–D Transient temperature and relative humidity at measured points Guizzardi et al. (2015) Masonry wall (external render + clay brick + cement mortar) Non–isothermal moisture transport 1–D/2–D Transient temperature and relative humidity at measured points, moisture content profile Ferroukhi et al. (2016) Red brick + polystyrene; red brick + plaster; chipboard + polystyrene; chipboard + plaster Non–isothermal moisture transport 1–D Temperature profile, relative humidity profile Medjelekh et al. (2016) Masonry wall (unfired clay brick + earth mortar) Non–isothermal moisture transport 1–D Transient temperature and relative humidity at measured points Sassine et al. (2017) Masonry wallette (clay brick, cement mortar) Thermal analysis 1–D Temperature profile Allam et al. (2018) Masonry wallette (clay brick, cement mortar) Non–isothermal moisture transport 1–D/2–D Temperature profile, relative humidity profile Delgado et al. (2019a) Ceramic brick; cement mortar; cement–lime mortar; Isothermal moisture transport 1–D Transient moisture mass; moisture content profile Calle et al. (2019) Masonry triplet (ceramic brick + natural hydraulic lime mortar) Isothermal moisture transport 1–D Moisture content profile X. Zhou et al., 2020) Masonry wallettes (brick + cement mortar) Isothermal moisture transport 1–D/2–D Moisture content profile Considering the possible causes for the existence of a hydraulic resistance at the brick–mortar interface, a commonly accepted explanation is related to the curing conditions of the bedding mortar between bricks (Groot, 1997; Brocken, 1998). In particular, bricks absorb water from the fresh mortar during application, which leads to a drop in the water–binder ratio of the mixture. Furthermore, water extraction from the mortar is also connected to the transport of fine binder particles towards the interface and a stratification of the mortar across the joint thickness. Consequently, one would expect a denser, more compact mortar at the interface. However, this is not always the case since a more porous mortar near the interface may be equally found in real scenarios. To account for this fact, Groot & Larbi (1999) hypothesized a reversed water flow from brick to mortar after compaction and initial hydration of the mortar, with subsequent Literature review 19 impact on the interface development. Overall, the quality of brick–mortar interface seems to be highly dependent on the curing history of the mortar. Derluyn et al. (2011) and Janssen et al. (2012) studied the absorption behaviour of brick–cement masonry composites with different interfacial configurations, namely perfect hydraulic contact (kaolin layer), and wet– and dry–cured specimens. Their studies confirmed that the interfacial effects were proportional to the water extraction from the mortar during curing. Further causes for the existence of a hydraulic resistance at the brick–mortar interface can be the presence of air gaps resulting from damage (cracks) or due to poor workmanship during the application of the fresh mortar (Groot & Gunneweg, 2010a). Even if the application and curing conditions were optimal, a certain hydraulic resistance is expected due to the pore structure discontinuity between the materials (Brocken, 1998). 2.3 HYGROTHERMAL BOUNDARY CONDITIONS Besides the characterisation of material properties, a comprehensive description of the boundary conditions is also necessary for a valid definition of the HT models. In Building Physics, these boundary conditions are usually classified into two groups according to the boundary location, namely indoor and outdoor. For indoor conditions, temperature and relative humidity are needed. For the outdoor conditions, temperature and relative humidity are equally necessary and depending on the complexity of the model, other data might be required as well, such as solar radiation, wind speed, precipitation, etc. Wind–driven rain is the largest source of moisture for any building structure (Karagiozis, 2001). However, it is a complex phenomenon, and the availability of data is still limited, so it is disregarded in many studies. The same applies to other phenomena commonly found in buildings, such as rising damp and roof leak. Besides the temperature and relative humidity, external and internal convective transfer coefficients (CTCs) are also necessary to define heat flux and moisture flux boundary conditions. The empirical determination of CTCs is a difficult task since they show dependency on the geometry of the studied surface as well as on local air flow, temperature and moisture conditions (Künzel, 1995). For engineering applications, a simplified approach is usually adopted. Thus, CTCs are assumed constant and estimated from analytical formulas or taken from tabulated values (Hagentoft et al., 2004; Defraeye et al., 2013). 2.4 HYGROTHERMAL MODELS Simultaneous heat and moisture transfer through porous media constitutes a highly coupled problem, that is, temperature balance and thermal material parameters vary with moisture content while the moisture–related properties and moisture equilibrium are dependent on the thermal distribution. Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 20 Comprehensive knowledge of each problem and their basic mechanisms is therefore necessary before tackling the combined phenomenon. The heat and moisture transport approaches presented hereafter were developed on the basis of the continuum model theory, which assumes that “matter is a hypothetical substance that is continuous throughout the spatial domain it occupies and can be described in that domain by a set of variables which are continuous and differentiable functions of the spatial coordinates and of time” (Bear & Bachmat, 1990). Under this assumption, a detailed definition of the exact microstructure of the material is not necessary. Instead, the internal structure of the material is averaged through Representative Elementary Volumes (REVs). In the case of porous media, the REV contains a representative configuration of the different phases, namely solid, liquid and gas (Figure 2.5). Figure 2.5. Schematic two–dimensional diagram of a non–saturated porous medium: representative elementary volume (REV) and its corresponding phases. Adapted from Pel (1995). 2.4.1 Heat problem Heat transfer is the process of energy exchange due to a temperature difference. There are three main types of heat transfer mechanisms, namely conduction, convection and radiation. Heat always flows from warmer to colder substances and the flow rate depends on the temperature difference, the area exposed and the type of material in between. Conduction may occur within gases, liquids or solids, although it is generally associated with heat transfer in solid materials. In a conductive process, the heat is transferred by the internal vibration of molecules or the movement of free electrons inside the material. Convection is caused by the movement of a fluid (gas or liquid), in which the fluid carries heat from one place to another; as the movement progresses, the hot fluid is replaced by cooler fluid. Convective heat transfer is defined as free or natural if buoyancy forces alone move the fluid, but it can also be induced by Solid phase Liquid phase Gas phase REV REV Liquid phase Solid matrix Air/vapour mixture Liquid island Literature review 21 mechanical means (forced convection). Thermal radiation occurs through electromagnetic waves and is related to the energy emitted or absorbed by a body as a result of being at a certain temperature. Heat conduction In a layer of solid material with each face at a different temperature, the heat flows from the side with higher temperature to the one with lower temperature. The heat flux is determined by Fourier’s Law, expressed for the one–dimensional case as: 𝑞𝑥=−𝜆 𝜕𝑇 𝜕𝑥 (2.3) where 𝑞𝑥 [W/m2] is the heat flux in 𝑥–direction, 𝜆 [W/(m·K)] is the thermal conductivity, and 𝑇 [K] is the temperature. The heat flux is proportional to the thermal conductivity of the material and the temperature difference and is inversely proportional to the thickness of the layer. Note that it is also common to find the heat conduction properties of building materials expressed in the form of thermal resistance, 𝑅𝑇=𝑙/𝜆 [m2·K/W], where 𝑙 [m] denotes the thickness of the material layer. In porous materials, heat fluxes may also be induced by moisture transport (Dufour effect). However, this effect has been proven negligible for the ranges of temperatures that are of interest for this research (Bažant & Thonguthai, 1987), and therefore it is not considered here. Heat convection The convective heat transfer between a surface and the environment is expressed by Newton’s Law of Cooling: 𝑞=ℎ𝑇 (𝑇𝑠𝑢𝑟𝑓−𝑇𝑒𝑛𝑣) (2.4) where ℎ𝑇 [W/(m2·K)] is the convective heat transfer coefficient. The determination of ℎ𝑇 is one of the main difficulties when dealing with this type of problem. Due to the lack of generalised predictive equations, ℎ𝑇 is generally estimated from empirical correlations as a function of wind speed (Azenha, 2009). Thermal radiation Radiation is a heat transfer phenomenon that affects all bodies by the fact of being at a certain temperature. Thermal agitations within the molecular structure of any material produce energy in the form of electromagnetic waves, which are emitted from the surface of the body, thus carrying the energy Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 22 away. Radiation is the only heat transfer mechanism that can take place in the vacuum, that is to say without a material medium (fluid or solid). Emissivity, 𝜀 [–], is the capacity of a body to emit radiation and it varies from 0 <𝜀< 1, with 𝜀= 1 as an ideal radiator or black body. The Stefan–Boltzmann law is used to determine the radiation emitted by a body: 𝐸=𝜀 𝜎 𝑇4 (2.5) where 𝐸 [W/m2] is the emitted radiation and 𝜎= 5.67E–8 W/(m2·K4) is the Stefan–Boltzmann constant. On the other hand, an object may also absorb radiation. The capacity of a body to absorb radiation is called absorptivity, 𝛼 [–], and analogously to 𝜀, it varies from 0 <𝛼< 1, being 𝛼= 1 the ideal black body. The net rate of radiation on a body can be calculated by the difference between the energy that is emitted by the surface and the irradiated energy being absorbed from the surroundings: 𝑞=𝜀 𝜎 𝑇𝑠𝑢𝑟𝑓4−𝛼 𝜎 𝑇𝑠𝑢𝑟𝑟4 (2.6) In the particular case of grey bodies, where 𝜀=𝛼, Eq. (2.6) becomes: 𝑞=𝜀 𝜎 (𝑇𝑠𝑢𝑟𝑓4−𝑇𝑠𝑢𝑟𝑟4) (2.7) For thermal problems circumscribed to civil engineering applications, only a certain range of wavelengths within the whole electromagnetic spectrum is of interest, namely ultraviolet, visible and infrared. Particular effects related to thermal radiation such as night–sky cooling, incident direct radiation from the sun (shortwave) and longwave radiation are of special interest for a detailed analysis of heat problems in civil engineering structures. For the sake or brevity, these phenomena will not be further discussed here. 2.4.2 Moisture problem The main moisture transport mechanisms in building materials are water vapour diffusion and liquid transport by capillarity. There is also bulk water vapour transport associated with airflow movements. However, the contribution of this factor to the overall moisture content is minor and it is therefore negligible (Scheffler, 2008). Similarly, the moisture fluxes induced by thermal gradients (Soret effect) can be disregarded (Janssen, 2011). Water vapour transport The main mechanism for water vapour transport in porous materials is vapour diffusion, which according to Fick’s First Law can be expressed as: Literature review 23 𝑔𝑣,𝑥=−𝐷𝑣 𝜕𝑐𝑣 𝜕𝑥 (2.8) where 𝑔𝑣,𝑥 [kg/(m2·s)] is the vapour flux in 𝑥–direction, 𝐷𝑣 [m2/s] is the vapour diffusivity or diffusion coefficient of vapour, and 𝑐𝑣 [kg/m3] is the vapour concentration. Diffusion may be described as a transport process proportional to a concentration gradient. If moist air is considered an ideal gas: 𝑝𝑣 𝑉=𝑚𝑣 𝑅𝑣 𝑇 (2.9) where 𝑝𝑣 [Pa] is the partial vapour pressure, 𝑉 [m3] is the volume, 𝑚𝑣 [kg] is the mass of vapour, and 𝑅𝑣= 461.5 [J/(kg·K)] is the universal gas constant for water vapour. The previous expression can be rearranged to describe the water vapour concentration: 𝑐𝑣=𝑚𝑣 𝑉=𝑝𝑣 𝑅𝑣 𝑇 (2.10) Therefore, the vapour diffusion may be expressed as: 𝑔𝑣,𝑥=− 𝐷𝑣 𝑅𝑣 𝑇 ·𝜕𝑝𝑣 𝜕𝑥 (2.11) By definition, relative humidity is expressed as: 𝜑= 𝑝𝑣 𝑝𝑣,𝑠𝑎𝑡(𝑇) (2.12) where 𝑝𝑣,𝑠𝑎𝑡(𝑇) [Pa] is the saturation vapour pressure dependent on temperature. Consequently, the vapour pressure may be defined as a function of relative humidity and temperature: 𝑝𝑣=𝜑 𝑝𝑣,𝑠𝑎𝑡(𝑇) (2.13) Different empirical formulas have been proposed for the definition of the saturation vapour pressure (Ochs et al., 2008). In this thesis, an adaptation of the expression by Murray (1967) presented in Monteith & Shatleworth (2013) will be used: 𝑝𝑣,𝑠𝑎𝑡= 610.7·107.5(𝑇−273.15 𝑇−35.85) (2.14) Considering Eq. (2.11), the vapour diffusion coefficient, the universal gas constant and the temperature are usually lumped together and expressed as ‘permeability’. Hence, the water vapour permeability of still air, 𝛿𝑎 [kg/(m·s·Pa)], is defined as follows: 𝛿𝑎= 𝐷𝑎 𝑅𝑣 𝑇 (2.15) Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 24 It must be noted that the definition of the vapour permeability of air is purely empirical and may be found with different expressions in the literature (Börjesson, 2013). In this thesis, the expression proposed by Schirmer (1938) will be used: 𝛿𝑎=2.31·10−5 𝑅𝑣·𝑇 𝑝0 𝑝(𝑇 273.15)1.81 (2.16) where 𝑝0= 101325 [Pa] is the standard atmospheric pressure, and 𝑝 [Pa] is the ambient barometric pressure. It is noted that Schirmer’s original formulation has been later adapted by other authors and it may appear in the literature with different forms. The vapour diffusion in porous materials is influenced by the pore structure of the material itself. In order to account for this fact, the vapour permeability of air is reduced by the so–called vapour diffusion resistance factor, 𝜇, which is characteristic of each material: 𝛿𝑣=𝛿𝑎 𝜇 (2.17) where 𝛿𝑣 [kg/(m·s·Pa)] is the water vapour permeability of the material. Due to the air pressure influence in Eq. (2.16), the vapour permeability of porous materials is sensitive to barometric changes. In particular, hygroscopic materials show increasing vapour permeability for lower air pressure values, whereas this effect is less noticeable for non–hygroscopic materials (M. Zhou et al., 2022). The vapour diffusion resistance factor, which represents the ratio of the permeability coefficients of water vapour in air and in the building material, may be further detailed as 𝜇=1/(𝜏·𝑎𝑉), where 𝜏 [–] is a tortuosity factor and 𝑎𝑉 [–] is the volume fraction of air–filled pores. The vapour resistance factor is independent of temperature (Tveit, 1966; McLean et al., 1990). Its dependence on the water content has been extensively discussed in the literature but a unified approach has not been achieved. An overall influence of the pore filling level seems reasonable since the increasing presence of liquid water reduces the pore space available for vapour diffusion (Scheffler, 2008). In effect, dry cup and wet cup test measurements usually result in different values of vapour permeability/resistance. On the other hand, some authors consider this influence negligible and treat the vapour transport coefficient as a constant (Künzel, 1995). Liquid water transport The liquid transport through a porous material may be explained by different approaches. The most common descriptions are the ones based on Fick’s Law of diffusion (diffusivity approaches) and the ones based on Darcy’s Law of permeability (conductivity approaches). Literature review 25 The diffusivity approaches make use of Fick’s First Law of diffusion and assume that liquid water movement follows a moisture concentration gradient such as: 𝑔𝑤,𝑥=−𝐷𝑤(𝑤)·𝜕𝑤 𝜕𝑥 (2.18) where 𝐷𝑤 [m2/s] is the liquid water diffusivity. It is known that 𝐷𝑤 varies with temperature and is strongly dependent on moisture content (Künzel, 1995). Moreover, liquid diffusivity is not a pure material property since it not only depends on the porous medium but on the boundary conditions as well (Krus, 1996). More specifically, 𝐷𝑤 is expected to change depending on the process, namely adsorption or desorption, and therefore it also presents hysteresis (Scheffler, 2008). On the other hand, conductivity approaches propose an adaptation of Darcy’s Law for unsaturated porous media (Galbraith, 1992) and use the capillary pressure gradient as the driving potential: 𝑔𝑤,𝑥=−𝐾𝑙(𝑤)·𝜕𝑝𝑐 𝜕𝑥 (2.19) where 𝐾𝑙 [kg/(m·s·Pa)] is the liquid water conductivity (also called liquid water permeability), which is a property of the porous medium and a function of the moisture content. Conductivity approaches may also use relative humidity as driving potential instead of the capillary pressure. Kelvin equation relates both parameters as follows: 𝑝𝑐=𝜌𝑤 𝑅𝑣 𝑇 𝑙𝑛𝜑 (2.20) By introducing Kelvin’s relation in Eq. (2.19) and after additional operations and simplificationsn, it is possible to arrive to: 𝑔𝑤,𝑥=−𝐷𝜑(𝜑)·𝜕𝜑 𝜕𝑥 (2.21) where 𝐷𝜑 [kg/(m·s)] is the liquid conduction coefficient, dependent on the porous medium and the relative humidity. Note that from Eqs. (2.18) and (2.21), the following connection between diffusivity and conductivity approaches can be established: 𝐷𝜑=𝜕𝑤 𝜕𝜑·𝐷𝑤=𝜉·𝐷𝑤 (2.22) where 𝜉=𝜕𝑤/𝜕𝜑 is the moisture storage capacity [kg/m3]. Moreover, the connection between diffusivity and conductivity approaches can be established through their respective liquid transfer coefficients such as: Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 26 𝐷𝑤(𝑤)=𝜕𝑝𝑐 𝜕𝑤·𝐾𝑙(𝑤)=𝒞·𝐾𝑙 (2.23) where 𝒞=𝜕𝑝𝑐/𝜕𝑤 is the derivative of the inverse moisture retention curve [m2/s2]. 2.4.3 Numerical modelling of heat and moisture transport For the analysis of coupled hygrothermal phenomena in masonry structures, it is necessary to define first a multi–physics model able to describe moisture diffusion and heat transfer in porous materials and multi–layered components. Several hygrothermal models for porous building materials are available in the literature. These models differ in their dimension (one–, two– or three–dimensional), the type of flow (steady–state, quasi–static or dynamic), the variable used to describe the moisture potential (moisture content, capillary pressure, relative humidity, or vapour pressure), the number of parameters needed as input information, as well as their assumptions and limitations. Temperature is universally accepted as driving potential for heat transport. However, moisture transport might be defined by different potentials, namely the moisture content, partial vapour pressure, relative humidity, hydraulic potential, or capillary pressure. In general, all these potentials can be related to one another and can be used equivalently with the proper transformations (Straube & Burnett, 2001). In addition to the chosen potential, the different models may be distinguished by their particular means of modelling the moisture component (Celia et al., 1990). Diffusivity approaches select a driving potential and lump all the transport mechanisms into a single moisture diffusivity equation. On the other hand, conductivity approaches separate vapour diffusion from liquid transport and model the flow as a parallel or series process. The first models dealing with the hygrothermal behaviour of porous materials were developed by (Philip & de Vries, 1957) and (Luikov, 1964). Their approaches rely on a set of coupled governing equations for heat and mass transfer with moisture content and temperature as dependent variables. However, these studies are focused on single–material cases and their direct application to multi–layered systems might present some problems: in composite structures made up of materials with dissimilar moisture sorption and diffusion properties, the moisture content is not continuous at one and the other side of the interface (Künzel, 1995). In order to overcome this obstacle, several authors have proposed hygrothermal models based on alternative moisture driving potentials, such as partial vapour pressure, e.g. Qin et al. (2009), Allam et al. (2018), or relative humidity, e.g. Khoshbakht et al. (2006), Lin et al. (2006), which verify the interfacial continuity between layers. In the field of Building Physics, these models Literature review 27 have been further extended to incorporate air into the mass transport and thus investigate the coupled heat, air and moisture (HAM) transfer, e.g. Tariku et al. (2010), Belleudy et al. (2016). Among the different approaches available in the literature, the hygrothermal model proposed by Künzel (1995) and extended by Künzel et al. (2001) is of especial interest for this research: it is a well– known approach applicable to multi–layered materials and uses transport coefficients that can be determined and validated through reasonably simple experimental tests. Künzel’s model considers heat transfer by conduction and enthalpy flow (phase change), such as: 𝜕𝐻 𝜕𝑡=∇·(𝜆 ∇𝑇)+𝐿𝑣 ∇ · (𝛿𝑣 ∇𝑝𝑣) (2.24) where 𝐻 [J/m3] is the enthalpy density, 𝜆 [W/(m·K)] is the thermal conductivity, 𝐿𝑣 [J/kg] is the latent heat of vaporization, 𝛿𝑣 [kg/(m·s·Pa)] is the water vapour permeability, and 𝑝𝑣 [Pa] is the partial vapour pressure. In turn, moisture transport consists of a two–phase diffusion model, such as: 𝜕𝑤 𝜕𝑡=∇·(𝐷𝑤 ∇𝑤)+ ∇ · (𝛿𝑣 ∇𝑝𝑣) (2.25) where 𝑤 [kg/m3] is the moisture content, and 𝐷𝑤 [m2/𝑠] is the liquid water diffusivity. For the liquid water diffusivity, Künzel (1995) proposes the following exponential expression: 𝐷𝑤=3.8·(𝐴𝑤 𝑤𝑐𝑎𝑝)2·1000(𝑤 𝑤𝑐𝑎𝑝 − 1) (2.26) where 𝐴𝑤 [kg/(m2·s0.5)] is the capillary absorption coefficient, and 𝑤𝑐𝑎𝑝 [kg/m3] is the capillary moisture content. It is noted that other diffusivity models have suggested different analytical expressions for the liquid water diffusivity, e.g. Pel et al. (1996), Krus & Holm, (1999), Carmeliet, Hens, et al. (2004), Carmeliet et al.(2007). The different expressions vary in their flexibility, complexity, and the number of required variables. For a more in–depth discussion of diffusivity approaches available in the literature, the reader is referred to Scheffler (2008). As mentioned in the section for the experimental works on multi–layered materials, a perfect contact is usually assumed for heat conduction through the interface. Consequently, the hygrothermal models commonly disregard any interfacial effects on heat transfer. On the other hand, the simulation of moisture transport in multi–layered cases still presents some challenges. Only a few investigations have focused on interface modelling for mass transfer problems in masonry, e.g. Brocken (1998), Derluyn et al. (2011), Calle et al. (2019), X. Zhou et al. (2020), and a unified approach to characterize the brick–mortar interfaces has not been reached yet. Moreover, the conclusions may vary from one author to another depending on the modelling technique and the eventual purpose of the simulation. For instance, as Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 34 This page has been intentionally left blank Materials, specimens, and experimental methods 35 CHAPTER 3 Materials, specimens, and experimental methods This chapter provides a brief overview of the studied materials, tested specimens and experimental methods used for material characterization. The chapter is organized in three different subsections. First, a general description of the chosen materials is provided. Subsequently, the configuration of the specimens used throughout the study is introduced. Finally, a detailed explanation of the different experimental procedures is given. The results obtained from the experimental tests as explained in this section will be introduced and discussed in the following chapter. Part of the information presented in this chapter has been published in Ramirez et al. (2021). 3.1 MATERIALS This section presents a description of the studied materials, namely fired–clay brick (B), Portland cement mortar (CM), natural hydraulic lime (NHL) mortar –both moulded (LM) and from masonry bed joints (LMJ)– and masonry composites. A selection of the studied material samples is shown in Figure 3.1. Figure 3.1. Selection of material samples: fired–clay brick, Portland cement mortar, natural hydraulic lime mortar (moulded and extracted from masonry bed joints), and masonry composites. The scale bar is 20 cm long. 3.1.1 Extruded fired–clay brick The masonry units selected for this study were commercial fired–clay solid extruded bricks. The extrusion process, as well as the drying and subsequent baking (or ‘firing’) were fully automated. According to the specifications of the manufacturer, the bricks were baked in a flash oven at 850 °C for 3 hours. Firing Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 36 temperatures below 1000 ℃ are considered low, and the resultant units are classified as ‘soft’ baked bricks (Fernandes et al., 2010). It must be noted that in this type of process, the actual baking temperature depends on the position of the units inside the oven, which may result in a certain level of variability between bricks of the same batch. An uneven heating of the brick mass is usually perceived within single units as well, which becomes apparent in the colour transition between the outer layer and the core (Figure 3.2). Clay particles become more sintered at higher baking temperatures, resulting in a lower porosity and a more intense red colour of the brick (Brocken, 1998). (a) (b) Figure 3.2. Cross–sections of fired–clay brick units: (a) transversal section through middle plane; (b) longitudinal section through middle plane. 3.1.2 Portland cement mortar Portland CEM I – 42.5R, as classified by the standard EN 197–1 (2011), was used as a reference for comparison with the lime–based mortar used in this study. The cement mortar was prepared with a binder–aggregate ratio 1:5 by volume and water–binder ratio 1:3 by weight. Standard moulds with dimensions 160 mm × 40 mm × 40 mm were used to cast the mixture. After casting, the mortar was kept in controlled curing conditions, 20 °C and 90 % RH, for 48 hours. Then, the specimens were demoulded and immersed in water at 20 °C for more than a year. 3.1.3 Natural hydraulic lime mortar The lime mixes analysed in this study were prepared using a commercial pre–mixed NHL–based mortar, NHL 3.5 (REABILITA Cal Consolidação). The mortar was prepared as suggested by the producer by blending 1 kg of the dry powder provided by the manufacturer with 0.15 kg of water. Standard moulds with dimensions 160 mm × 40 mm × 40 mm were used to cast the mixture. For moulded specimens, the freshly cast mortar was kept for 48 hours in controlled curing conditions, 20 °C and 90 % RH. Then, the specimens were demoulded and maintained in laboratory conditions, 20 °C and 60 % RH, for more than a year. Materials, specimens, and experimental methods 37 3.1.4 Masonry Masonry samples (brick + mortar) were extracted from a wall using a diamond coring wet drilling machine. The wall was constructed using the same bricks and NHL–based mortar discussed above. The bricks were docked in water before placement to prevent undesired suction of water from the fresh mortar. The average thickness of the mortar joints was 12 mm. Right after the wall was built, it was covered with a polyethylene sheet for 48 hours to prevent water evaporation. After that, the cover was removed and the wall was kept in laboratory conditions, 20 °C and 60 % RH, for more than a year before the cores were extracted. Examples of the studied masonry specimens are shown in Figure 3.3. (a) (b) Figure 3.3. Selection of masonry specimens: (a) cylinders for drying tests; (b) cylinders for capillary absorption tests. 3.2 SPECIMENS Fired–clay extruded bricks were used in different formats depending on the corresponding test setup. Initially, whole brick units with nominal size 200 mm × 100 mm × 50 mm were tested. Subsequently, the external surfaces of the units (5 mm) were ground, and the resulting prisms were cut into smaller pieces for further testing (Figure 3.4). Figure 3.4. Cutting layout of brick units and mortar prisms (average dimensions in mm). Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 38 As for the mortars, standard moulds with dimensions 160 mm × 40 mm × 40 mm were used to cast the different mixes. Initially, whole mortar prisms were used. Then, the prisms were cut into smaller pieces for further testing (Figure 3.4). Additionally, lime mortar discs extracted from masonry bed joints (Figure 3.5) were analysed to determine differences with respect to the moulded counterpart. The multi–layered specimens extracted from the masonry wall consisted of cylinders with different stacking arrangements, i.e. different B+LMJ configurations (Figure 3.5). The average diameter of the cylinders was Ø= 51.30 mm. Figure 3.5. Lime mortar discs and multi–layered specimens extracted from masonry wall (schematic representation) (average dimensions in mm). The number and type of specimens used for the characterization of single materials are summarized in . Additionally, the number and configuration of masonry specimens used for the characterization of multi– layered materials are summarized in Table 3.1. Table 3.1. Number of single–material specimens tested. Material Specimen Test and test direction (dimensions in mm) Vacuum saturation Immersion at atmospheric pressure IRA Sorption isotherm Capillary absorption 1–D drying test Vapour permeability CTE CHE MIP X Y Z X Y Z Z X X B 200x100x50±2 10 10.(1) 10.(1) – 5.(2) – – – – – – – 40x40x40±1 15.(3) – – – 5.(3) 5.(3) 5.(3) 5.(4) 5.(4) 5.(4) – – – – 50x50x20±1 – – – – – – – – – – 5.(3) – – – 30x30x10±1 – – – 5.(3) – – – – – – – – – 10.(5) 160x40x40±0.5 – – – – – – – – – – – 6 6.(6) – CM 160x40x40±0.5 4 4.(1) – – 4.(1) – – – – 4.(1) 4.(6) – 40x40x40±0.5 3.(7) – – – 3.(1) – – 3.(4) – – – – – – 30x30x10±1 – – – 3.(7) – – – – – – – – – 2.(5) LM 160x40x40±0.5 4 4.(1) – – 4.(1) – – – – 4.(1) 4.(6) – 40x40x40±0.5 3.(7) – – – 3.(1) – – 3.(4) – – – – – – 30x30x10±1 – – – 3.(7) – – – – – – – – – 2.(5) LMJ ∅51.3, h= 12±1 5 – – 5 – – – – – – – – – 4.(5) Note: IRA is the initial rate of absorption; CTE is the coefficient of thermal expansion; CHE is the coefficient of hygric expansion; MIP Is Mercury Intrusion Porosimetry .(1) Same specimens used for vacuum saturation .(2) 5–brick unit subset chosen from the initial 10 brick units tested for vacuum saturation, immersion at atmospheric pressure and IRA .(3) Cut from 5–brick unit subset .(4) Same specimens used for water absorption .(5) Small pieces cut from specimens used for sorption isotherm .(6) Same specimens used for CTE .(7) Cut from mortar prism Materials, specimens, and experimental methods 39 Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 40 Table 3.2. Number of masonry specimens tested. Masonry Specimen Test and test configuration (dimensions in mm) Capillary absorption 1–D drying test +Z −Z Drying from brick Drying from mortar M1 Cylinder ∅51.3 hB= 45±1, hLMJ= 12±1 8 8 (1) – – M2 Cylinder ∅51.3 hB= 45±1, hLMJ= 12±1 8 8 (1) – – M4 Cylinder ∅51.3 hB= 48±1, hLMJ= 12±1 6 – – – D1 Cylinder ∅51.3 hB= 22±2, hLMJ= 12±1 – – 6 6 Note: hB is the height of the brick; hLMJ is the height of the mortar joint; Z is the direction perpendicular to the bed joint and bed face of the brick (1) Same specimens tested in the opposite direction 3.3 EXPERIMENTAL METHODS Standard experimental procedures were followed in order to determine the material properties needed for a comprehensive hygro–thermo–mechanical characterization. The focus of the experimental studies was set on physical and moisture–related properties. Additionally, thermo– and hygro–mechanical properties were also studied to understand the material behaviour with respect to temperature and moisture changes. Finally, mercury intrusion porosimetry (MIP) tests were conducted to define the micro– structure of the materials. Examples of the experimental setups are shown in Figure 3.6. 3.3.1 Oven–drying for pre–conditioning of samples Most experimental methods applied in the current study required an initial pre–drying of the materials. Such initial pre–conditioning was performed by means of oven–drying in accordance with ISO 12570 (EN ISO 12570:2000 + A1:2013, 2000). Oven–drying at 105°C was used for fired–clay bricks. Conversely, the drying temperature was reduced to 70 °C for mortar and masonry specimens to avoid microstructural damage or removal of chemically bound water from the cementitious matrix (Feng et al., 2013). The drying process was concluded when the change of mass between two consecutive weight measurements with a difference of at least 24 hours was less than 0.1 % of the total mass of the specimen. Materials, specimens, and experimental methods 41 (a) (b) (c) (d) (e) (f) Figure 3.6. Examples of test setups used to determine material properties: (a) vacuum saturation; (b) immersion at atmospheric pressure; (c) capillary absorption; (d) cup test; (e) coefficient of thermal expansion; (f) coefficient of hygric expansion. 3.3.2 Vacuum saturation tests Vacuum saturation tests were performed following the recommendations of RILEM TC 25–PEM (1980) to determine open porosity, 𝜙𝑜 [–], bulk density, 𝜌𝑏𝑢𝑙𝑘 [kg/m3], and saturation moisture content, 𝑤𝑠𝑎𝑡 [kg/m3]. The studied samples comprised brick units, mortar prisms, brick and mortar cubes, and bed joint mortar discs (Table 3.1). Due to the high variability usually shown by fired–clay bricks, a sample of 10 randomly chosen units belonging to the same batch was analysed initially. Then, a subset of 5 statistically comparable specimens was selected for further studies. For mortar specimens, a sample of 4 prisms for each mortar type and 5 bed joint mortar discs were used. Initially, the specimens were oven–dried following the procedure previously explained (see Section 3.3.1). The samples were left to cool down and then placed in an evacuation vessel where the pressure was lowered below 100 mbar to remove the air from the open pores of the material. The vacuum pressure was maintained for 4 hours. Subsequently, tap water at 15–20 °C was gradually introduced into the vessel until the water level was 1–2 cm above the specimens. Vacuum was maintained during the Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 42 introduction of water and for the following 4 hours. Subsequently, the system was set back to atmospheric pressure and the specimens were left under water for 24 hours. Finally, the immersed mass, 𝑚𝑖𝑚 [kg], was determined by weighing under water (hydrostatic weighing). Then, the specimens were wiped with a dampened cloth and weighed in air to determine the saturated mass, 𝑚𝑠𝑎𝑡 [kg]. The bulk volume, 𝑉𝑏𝑢𝑙𝑘 [m3], was calculated from the masses measured in air and under water: 𝑉𝑏𝑢𝑙𝑘=𝑚𝑠𝑎𝑡−𝑚𝑖𝑚 𝜌𝑤 (3.1) where 𝜌𝑤 [kg/m3] is the density of water, assumed 1000 kg/m3 in normal conditions. Consequently, the bulk density, 𝜌𝑏𝑢𝑙𝑘 [kg/m3], was calculated as: 𝜌𝑏𝑢𝑙𝑘=𝑚𝑑𝑟𝑦 𝑉𝑏𝑢𝑙𝑘 (3.2) Similarly, the open porosity, 𝜙𝑜 [–], was calculated as: 𝜙𝑜=𝑚𝑠𝑎𝑡−𝑚𝑑𝑟𝑦 𝑚𝑠𝑎𝑡−𝑚𝑖𝑚 (3.3) Saturation moisture content, 𝑤𝑠𝑎𝑡 [kg/m3], was derived from open porosity as: 𝑤𝑠𝑎𝑡=𝜙𝑜·𝜌𝑤 (3.4) The total porosity, i.e. open + closed porosity, was not studied since the volume of closed pores does not take part in moisture transport. 3.3.3 Immersion at atmospheric pressure Immersion tests were performed according to EN 772–21 (2011). This type of tests is meant to provide water absorption, 𝑊𝑠 [%] or 𝑊𝑔,𝑠 [kg/kg], and is exclusively focused on masonry units. In the present study, however, immersion tests were performed on mortar prisms as well. For all cases, the main guidelines defined by the standard procedure were followed. Initially dried specimens were placed in a tank with water at room temperature. After 24 hours immersion at atmospheric pressure, the specimens were taken from the tank, the excess of water removed with a damp cloth, and their mass was recorded. In the context of the present work, the main purpose of this procedure was to study the possibility of identifying a fixed capillary moisture content. Therefore, after the first measurement, the specimens were placed back in the tank and the procedure was repeated every 24 hours for several days. Materials, specimens, and experimental methods 43 3.3.4 Initial rate of absorption The initial rate of water absorption, IRA, is a parameter used in masonry as a measure of brick suction. It represents the weight of water absorbed in 1 minute by the bed face of the brick when immersed in a shallow water basin, 5±1 mm. The IRA may range from below 1 kg/(m2·min) for low suction rate bricks up to 3–4 kg/(m2·min) for high suction rate bricks (Groot & Larbi, 1999). The procedure for the initial rate of absorption test is established in EN 772–11 (2011). The tests were performed inside a climatic chamber with controlled temperature and humidity conditions of 20 ℃ and 60 % RH. The water used for the tests was conditioned to room temperature beforehand to avoid temperature–dependent phenomena associated with the change of viscosity of the water (Feng & Janssen, 2016). IRA was studied on the initial 10 brick units tested for vacuum saturation and immersion at atmospheric pressure. 3.3.5 Static gravimetric tests The sorption isotherms (adsorption/desorption) of the materials were obtained through static gravimetric tests according to EN ISO 12571 (2013). The sorption isotherms represent the moisture storage capacity of a material in the hygroscopic range, namely from dry state, i.e. 0 % RH, to 93–95 % RH. In the over– hygroscopic or capillary range, i.e. above 93–95 % RH, the sorption isotherms become extremely steep and slight variations in relative humidity result in large changes in moisture content. Thus, the higher part of the moisture storage curve must be completed by other means, namely pressure plate experiments or results derived from mercury porosimetry (Krus, 1996). The size of the specimens used for these tests (see Table 3.1) was chosen following the recommendations by Feng et al. (2013), who proved that smaller specimens help speed up the process without compromising the accuracy. Brick and mortar specimens were cut from the brick units and mortar prisms used for the previous tests (Figure 3.4): 1 brick piece from each unit (5 replicates in total) and 3 mortar pieces from a single prism (3 replicates for each type of mortar). Additionally, lime mortar discs extracted from masonry bed joints (5 replicates) were tested as well. To define the sorption isotherms, the equilibrium moisture content of the materials was determined at different relative humidity levels. It must be noted that the moisture content is usually expressed either as mass of adsorbed water per volume of dry material, 𝑤 [kg/m3], or as mass of adsorbed water per mass of dry material, 𝑤𝑔 [kg/kg]. The latter format, known as gravimetric moisture content, is used for the static gravimetric tests. Nonetheless, both expressions are related through: 𝑤=𝑤𝑔·𝜌𝑏𝑢𝑙𝑘 (3.5) Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 50 deformation. The specimens were measured in their longitudinal direction using a digital dial gauge in a fixed frame (see Figure 3.6f). The coefficient of hygric expansion, 𝛼ℎ [m3/kg] or 𝛼ℎ,𝑔 [kg/kg], was determined for each exposure step as: 𝛼ℎ=𝜀ℎ𝑖+1 −𝜀ℎ𝑖 𝑤𝑖+1−𝑤𝑖 (3.12) where 𝜀ℎ [–] is the strain caused by the moisture content difference between two consecutive steps, ∆𝑤 [kg/m3], and 𝑤 [kg/m3] is the moisture content (which may be expressed as gravimetric moisture content, 𝑤𝑔 [kg/kg], as well). Finally, an analytical expression for the coefficient of hygric expansion, i.e. 𝛼ℎ(𝑤), was derived from the slope of the curve relating hygric strains and moisture content: 𝛼ℎ(𝑤)=𝑑𝜀ℎ 𝑑𝑤 (3.13) 3.3.11 Mercury intrusion porosimetry The pore size distribution of the materials was determined by mercury intrusion porosimetry (MIP) according to ASTM D4404–18 (2018). The procedure is derived from fluid behaviour in an unsaturated medium assuming that a fluid intrusion, in this case mercury, is dependent on pressure. Hence, mercury is forced into the pores of the material following incremental pressure steps, and the relation between intruded mercury content and applied pressure is recorded. Assuming a cylindrical pore model (Krus, 1996), the Young–Laplace equation defines the pressure at which a pore fills (or empties) according to its pore opening size: 𝑝𝑐=2𝜎 𝑟· 𝑐𝑜𝑠 (𝜃) (3.14) where 𝜎 [N/m] is the surface tension, 𝑟 [m] is the pore radius, and 𝜃 is the contact angle. Therefore, knowing the pressure applied at each level and with the appropriate values of surface tension and contact angle for mercury, it is possible to obtain the effective capillary size and consequently the pore size distribution (Figure 3.9). It is noted that MIP can provide relevant information about the microstructural characteristic of the materials within the macro–pore and meso–pore ranges, namely 2–50 nm and >50 nm, respectively, according to the pore size ranges defined by IUPAC (International Union of Pure and Applied Chemistry) (IUPAC, 1972). However, for the lower meso–pore and micro–pore ranges, different methods must be used, e.g. physisorption. Materials, specimens, and experimental methods 51 Figure 3.9. Relation between pore size, capillary pressure, and relative humidity according to Young–Laplace and Kelvin equations, together with the classification of pores according to IUPAC and the measuring range of the experimental methods. Adapted from Brocken (1998). For the current study, the tests were performed using a mercury porosimeter equipment (Micromeritics AutoPore IV 9500 V1.07) with a range of applied pressure between 32990 psia (227.46 MPa) and 0.53 psia (3.65 kPa), corresponding to minimum and maximum pore sizes of 0.005 μm and 345 μm, respectively. The contact angle for the intruding mercury meniscus was defined as 130° (Ma, 2014). In addition, mercury surface tension and mercury density were assumed 0.485 N/m and 13.5335 g/ml, respectively. The specimens for MIP were taken from the samples previously used for the determination of sorption isotherms. From each material specimen, 2 pieces were cut for MIP testing and the obtained results were averaged. The resulting pieces had a mass ranging from 0.90 g to 1.95 g. Before the test, the specimens were stored in a glass desiccator with silica gel until dry state was attained. Finally, with the appropriate transformations for the values of surface tension and contact angle, MIP results can be transformed into the (desorption) moisture retention curve. In this regard, the surface tension and contact angle for liquid water were defined as 0.072 N/m and 0°, respectively. Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 52 This page has been intentionally left blank Experimental results 53 CHAPTER 4 Experimental results The current chapter presents the results of the material characterization and experimental testing procedures introduced in the previous chapter. The chapter is divided into two main subsections. The first section is dedicated to the experimental results obtained for single constituent materials. In turn, the second part is devoted to the hygric tests performed on multi–layered masonry specimens. Relevant discussions of the experimental procedures and the obtained results are given in the corresponding section. The results presented herein will be used as input and validation data for the moisture transport simulations as explained in the following chapter. Part of the information presented in this chapter has been published in Ramirez et al. (2021). 4.1 RESULTS ON SINGLE CONSTITUENT MATERIALS The results obtained from experimental tests in mono–material specimens are presented herein. The experimental campaign comprised vacuum saturation, immersion at atmospheric pressure, gravimetric tests to determine the moisture storage curves, cup tests, initial rate of absorption, capillary absorption, drying tests, tests to determine the coefficients of thermal expansion and hygric expansion, and mercury intrusion porosimetry. 4.1.1 Vacuum saturation tests results The properties derived from vacuum saturation tests are collected in Table 4.1. In the table, CoV indicates the coefficient of variation. Vacuum saturation tests performed on whole bricks and on brick cubes extracted from those initial units produced the same results in terms of bulk density and open porosity. Similarly, no difference was observed between mortar prisms and mortar cubes cut out of them. Therefore, the use of smaller specimens is justified and is recommended to speed up the process and guarantee the saturation of the innermost spaces. As it was expected, the properties of the lime mortar obtained from masonry bed joints differed from those obtained from moulded specimens even though the mixes were prepared with the same composition and in the same controlled environment. Overall, LMJ showed higher bulk density and lower open porosity, and consequently lower saturation moisture content. Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 54 Table 4.1. Results from vacuum saturation tests (CoV between parentheses). Material 𝜌𝑏𝑢𝑙𝑘 𝜙𝑜 𝑤𝑠𝑎𝑡 [kg/m3] [–] [kg/m3] Brick (B) 1900 (2.69) 0.280 (6.76) 280 (6.76) Cement mortar (CM) 2000 (0.15) 0.210 (1.05) 210 (1.05) Lime mortar (LM) 1990 (0.81) 0.255 (2.53) 255 (2.53) Lime mortar joints (LMJ) 2060 (1.32) 0.230 (6.93) 230 (6.93) 4.1.2 Immersion at atmospheric pressure The results of the immersion tests at atmospheric pressure are presented in Figure 4.1. The graphs show the evolution of gravimetric moisture content, 𝑤𝑔 [kg/kg], with immersion time for brick units (Figure 4.1a) and mortar prisms (Figure 4.1b). As defined in EN 772–21 (2011), 𝑊𝑔,𝑠 [kg/kg] is determined for clay bricks after 24 hours of immersion at atmospheric pressure. Thus, the average value for the studied units 𝑊𝑔,𝑠= 0.128 kg/kg. However, it must be noted that the bricks continued absorbing water with prolonged immersion time. This phenomenon is expected due to air entrapment in the pore structure during the imbibition process (Descamps, 1997). Thus, the saturation level of bricks at 24 hours was around 97.3 % the level of saturation attained after 72 hours of immersion. Conversely, the mortar specimens had reached ca. 99 % the final saturation already after the prescribed period of 24 hours. The bigger volume of masonry units is thought to be the reason for the slower saturation process observed in the brick material. In other words, air entrapment is more evident in bigger specimens. 4.1.3 Initial rate of absorption The average initial rate of absorption (IRA) obtained for brick units was 0.30 kg/(m2·min), with a coefficient of variation CoV = 15.42 %. According to the classification by Groot & Larbi (1999), this value indicates that the studied fired–clay bricks belong to the low suction rate category, i.e. IRA < 1 kg/(m2· min). The high CoV reveals a certain scatter within the tested sample, which can be related to the presence of defects or irregularities in the outermost layer of the brick units. Moreover, the short duration of IRA tests might exacerbate any difference that could otherwise stabilize with prolonged contact with water. Experimental results 55 (a) (b) Figure 4.1. Results of the immersion test at atmospheric pressure: (a) brick units; (b) cement mortar (CM) and lime mortar (LM) specimens. 4.1.4 Moisture isotherm The data collected from static gravimetric tests are shown in Figure 4.2. The results showed clear similarities between brick and lime mortar, whereas cement mortar and lime mortar from joints followed a different trend. On one hand, B and LM hardly showed moisture adsorption until high RH values, revealing a low hygroscopic response. Moreover, no significant differences were found between adsorption and desorption. Conversely, hysteresis was observed in CM and LMJ. CM showed considerable moisture adsorption for lower RH levels, confirming a markedly hygroscopic nature. Likewise, LMJ showed moisture adsorption for lower RH levels, and therefore a more hygroscopic response than its moulded counterpart. The experimental measurements were fitted with analytical expressions found in the literature. In particular, the model proposed by Künzel (1995) was used to describe the sorption behaviour of B and LM. For CM and LMJ, the model established by Mualem (Mualem, 1976) was adopted. The expression proposed by Künzel can be written as: 𝑤(𝜑)=𝑤𝑐𝑎𝑝·(𝜓−1)·𝜑 𝜓−𝜑 (4.1) where 𝑤(𝜑) [kg/m3] is the moisture content as a function of relative humidity, 𝜑 [–], 𝑤𝑐𝑎𝑝 [kg/m3] is the capillary moisture content, and 𝜓 [–] is a fitting parameter. Künzel’s model stems from a simplified form of the BET equation (Brunauer et al., 1938) and is applicable to sorption curves with a marked exponential trend, i.e. non–hygroscopic, capillary–active materials. 012 24 36 48 60 72 0.00 0.04 0.08 0.12 0.16 Gravimetric moisture content [kg/kg] Time [h] 012 24 36 48 60 72 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Gravimetric moisture content [kg/kg] Time [h] LM CM Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 56 (a) (b) (c) (d) Figure 4.2. Static gravimetric tests results and fitting curves for the sorption isotherms of the studied materials at 23 °C: (a) brick (B), and (b) lime mortar (LM), fitted with Künzel’s model (Künzel, 1995); (c) cement mortar (CM), and (d) lime mortar from masonry joints (LMJ), fitted with Mualem’s model (Mualem, 1976). On the other hand, the model proposed by Mualem is extensible to hygroscopic materials. This model is a constrained form of the more generalized equation proposed by van Genuchten (1980): 𝑤(𝑝𝑐)=𝑤𝑐𝑎𝑝·[1+(𝑎·𝑝𝑐)𝑛]−𝑚 (4.2) where 𝑤(𝑝𝑐) [kg/m3] is the moisture content as a function of capillary pressure, 𝑝𝑐 [Pa], and 𝑎 [1/Pa], 𝑛 [–] and 𝑚 [–] are fitting parameters. It is noted that capillary pressure and relative humidity can be related through the Kelvin equation, such as: 𝑝𝑐=𝜌𝑤·𝑅𝑣·𝑇·𝑙𝑛𝜑 (4.3) where 𝜌𝑤 [kg/m3] is the density of water, 𝑅𝑣 [J/(kg·K)] is the universal gas constant for water vapour, and 𝑇 [K] is the absolute temperature. 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Relative humidity,  [-] Gravimetric moisture content [kg/kg] Adsorption Desorption Adsorption Desorption 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Relative humidity,  [-] Gravimetric moisture content [kg/kg] Adsorption Desorption Adsorption Desorption 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Relative humidity,  [-] Adsorption Desorption Adsorption Desorption Gravimetric moisture content [kg/kg] 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Relative humidity,  [-] Gravimetric moisture content [kg/kg] Adsorption Desorption Adsorption Desorption Experimental results 57 Considering van Genuchten’s expression, Mualem fixed 𝑚=1−1/𝑛 (Mualem, 1976): 𝑤(𝑝𝑐)=𝑤𝑐𝑎𝑝·[1+(𝑎·𝑝𝑐)1/(1−𝑚)]−𝑚 (4.4) The curves obtained from these analytical expressions are plotted in Figure 4.2 together with the experimental points. The corresponding fitting parameters for the different cases are presented in Table 4.2 together with relevant criteria to evaluate the accuracy of the model. In this case, the accuracy was determined by the adjusted coefficient of determination (𝑅𝑎𝑑𝑗 2) and the normalised root mean square error (%NRMSE) between predicted and measured points. The adjusted 𝑅2 takes into account the effect of the number of fitting parameters and is given as: 𝑅𝑎𝑑𝑗 2=(𝑁−1)𝑅2−(𝑀−1) 𝑁−𝑀 (4.5) where 𝑁 is the number of measured data and 𝑀 is the number of fitting parameters. The %NRMSE is expressed as a percentage and is defined as follows: %NRMSE= 1 ∑𝑥𝑒𝑥𝑝,𝑖/𝑁 𝑁 𝑖=1 √1 𝑁∑(𝑥𝑒𝑥𝑝,𝑖−𝑥𝑝𝑟𝑒,𝑖)2 𝑁 𝑖=1 (4.6) where 𝑥exp and 𝑥pre are the measured values from the experiments and the predicted values from the fitted curves, respectively. A lower value of %NRMSE indicates a better match. Table 4.2. Fitting parameters and accuracy of the modelled sorption isotherms. Material Künzel (Eq. (4.1)) Mualem (Eq. (4.4)) 𝑅𝑎𝑑𝑗 2 %NRMSE B 𝜓𝑎𝑑𝑠= 1.0055 – 0.999 2.30% 𝜓𝑑𝑒𝑠= 1.0070 – 0.999 2.49% LM 𝜓𝑎𝑑𝑠= 1.0052 – 0.999 5.30% 𝜓𝑑𝑒𝑠= 1.0066 – 0.999 4.49% CM – 𝑎𝑎𝑑𝑠 = 1.43E–6 [1/Pa] 𝑚𝑎𝑑𝑠= 0.285 0.989 8.26% – 𝑎𝑑𝑒𝑠 = 2.51E–6 [1/Pa] 𝑚𝑑𝑒𝑠= 0.213 0.982 8.10% LMJ – 𝑎𝑑𝑒𝑠 = 3.80E–6 [1/Pa] 𝑚𝑑𝑒𝑠= 0.352 0.990 11.63% – 𝑎𝑑𝑒𝑠 = 1.19E–5 [1/Pa] 𝑚𝑑𝑒𝑠= 0.235 0.999 2.42% Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 58 4.1.5 Vapour permeability The results from dry cup and wet cup tests performed on brick specimens gave 𝜇𝑑𝑟𝑦= 34.14 (CoV = 12.64 %) and 𝜇𝑤𝑒𝑡= 13.50 (CoV = 17.85 %), respectively. Vapour permeability was only tested in the bed direction (Z) since the cutting of the original units after removal of the outermost layer (5 mm) did not allow to obtain the specimens for the other directions (Figure 3.4). It must be noted that the results obtained from cup measurements showed the highest scatter among the tests performed in this study. A low reliability for this type of test has been indicated before, see e.g. Roels et al. (2004), Feng et al. (2015), Hens (2016), Feng et al. (2020). The water vapour resistance factor, 𝜇, indicates how many times lower the vapour diffusion in the material is in comparison with vapour diffusion in still air. By definition, 𝜇= 1 for stagnant air. Expressing the water vapour permeability through the vapour resistance factor by means of Eq. (3.6) has the advantage of assigning the temperature and pressure dependencies of water vapour diffusion to the empirical term 𝛿𝑎 (𝑇,𝑝), so that the vapour resistance factor is independent of these variables and becomes a constant characteristic for each material. Nonetheless, as the values for our brick specimens show, vapour diffusion tests performed at different relative humidity levels, e.g. dry and wet cups, usually result in different permeability values for the same material (Kumaran, 1996; Roels, Carmeliet, et al., 2003). Consequently, vapour diffusion (expressed through either vapour permeability or vapour resistance factor) must be a function of moisture content. There have been considerable efforts in the literature to explain this phenomenon, e.g. Künzel (1995), Krus (1996), Scheffler (2008), but a univocal standpoint has not been reached. Most authors agree on the overall influence of the pore–saturation level since the presence of water islets in the pore system modifies the effective space available for vapour diffusion (Scheffler, 2008). However, the debate appears when defining the effect of moisture condensation in the pores for high moisture contents and subsequent reduction of the space accessible to vapour transport. Künzel (1995) argued that vapour diffusion might be either obstructed or (this is the most common assumption) enhanced through the water islets depending on the local conditions of temperature and moisture content inside the material. On the other hand, Krus (1996) introduced the phenomenon of surface diffusion, i.e. advective water transport along the adsorbed liquid film on pore walls, to explain the differences observed experimentally between dry and wet cup tests. Surface diffusion becomes noticeable at high humidity levels, but in practice, it cannot be distinguished from vapour transport and therefore it is lumped together with diffusion, which could cause the observed differences in apparent vapour permeability. The additional liquid transport superimposed on diffusion would be likely to increase the observed vapour permeability, therefore providing a lower resistance value for wet cup Experimental results 59 measurements. Indeed, this is in agreement with the observations made for the brick specimens in our study. 4.1.6 Capillary absorption Water uptake tests for material characterization were performed on whole units, brick cubes and mortar prisms along different directions. The results are summarized in Table 4.3. Table 4.3. Results from water absorption tests (CoV between parentheses). Material 𝐴𝑤,𝑋 𝐴𝑤,𝑌 𝐴𝑤,𝑍 𝐴𝑤.(1) (2) 𝑤𝑐𝑎𝑝.(3) [kg/(m2·s0.5)] [kg/(m2·s0.5)] [kg/(m2·s0.5)] [kg/(m2·s0.5)] [kg/m2] B units 0.118 (8.43) 0.090 (5.51) 0.068 (7.36) – 250 (2.35) B cubes 0.104 (6.31) 0.089 (7.78) 0.061 (5.86) – 240 (1.07) LM 0.237 (1.62) 0.234 (1.59) 0.233 (1.46) 0.235 (1.57) 225 (2.05) CM 0.059 (3.96) 0.059 (5.32) 0.060 (7.77) 0.059 (6.49) 180 (2.45) .(1) Isotropic .(2) Sample of 4 moulded prisms .(3) Average of all tests The capillary moisture content, 𝑤𝑐𝑎𝑝 [kg/m3], was determined as the average moisture content at the end of the water absorption tests. It should be noted that the capillary moisture content is always lower than the saturation moisture content defined from open porosity, i.e. 𝑤𝑐𝑎𝑝<𝑤𝑠𝑎𝑡, due to air entrapment in the pore structure during imbibition (Descamps, 1997). For prolonged imbibition or immersion times, 𝑤𝑐𝑎𝑝 will approach 𝑤𝑠𝑎𝑡 as the air trapped in the pores dissolves into the water (Scheffler, 2008). This phenomenon was already pointed out in the specimens subjected to immersion at atmospheric pressure (see Section 4.1.2), which exhibited a slow, progressive saturation process. Thus, it can be concluded that 𝑤𝑐𝑎𝑝 is more a fuzzy limit than a fixed value. For the studied materials, the comparison between saturation and capillary moisture contents gave 𝑤𝑐𝑎𝑝≈ 80 % – 90 % 𝑤𝑠𝑎𝑡, which is an indication of the quantity of pores that can be classified as capillaries with respect to the overall open porosity. The results obtained for whole bricks are shown in Figure 4.3 for the three studied directions: (a) header or extrusion direction, labelled X; (b) stretcher direction, labelled Y; (c) bed direction, labelled Z. Note that each set of tests ended up in a different plateau since the exposed surface was not the same Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 66 4.1.9 Coefficient of hygric expansion The results for the coefficient of hygric expansion (CHE) of the studied materials are presented in Figure 4.8, together with the fitted curves representing the CHE as a function of moisture content. It is noted that the CHE changes considerably for lower moisture contents, whereas the variation is small for higher moisture contents. The CHE obtained for B (Figure 4.8a) is much higher than the ones calculated for LM and CM (Figure 4.8b), which are in the same range and similar to one another. All the studied cases are well described by a power type (allometric) function with respect to the gravimetric moisture content. (a) (b) Figure 4.8. Coefficient of hygric expansion and fitting curves for the studied materials: (a) brick (B); (b) lime mortar (LM) and cement mortar (CM). In the literature, the CHE can appear as a linear relation, such as 𝜀ℎ=𝛽(𝑤−𝑤0). In this context, the CHE is commonly known as coefficient of hygroscopic swelling (CHS). The linear assumption is generally valid to describe the material behaviour in a certain portion of the hygroscopic range. However, the same simplification may prove faulty for higher moisture contents and therefore its application is not recommended for cases dealing with the whole moisture range. For reference, the CHS values obtained for the different materials analysed in this study are presented hereunder, expressed in strain per 1 % moisture mass gain: 𝛽𝐵= 0.0520, 𝛽𝐿𝑀= 0.0098, and 𝛽𝐶𝑀= 0.0048. These values were calculated as the slope of the linear regression curve relating the hygric strains and moisture contents calculated for 55, 80 and 90 % RH. 4.1.10 Mercury intrusion porosimetry The mercury intrusion porosimetry (MIP) results for the studied materials are shown in Figure 4.9, Figure 4.10 and Figure 4.11, in terms of cumulative porosity and percentage of porous volume occupied by each pore size, percentage of pores with respect to pore size, and differential intruded volume as a 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.00 0.05 0.10 0.15 0.20 0.25 Gravimetric moisture content, wg [kg/kg] h,B = 1.40·10−4 / w 0.72 g Coefficient of hygric expansion 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.00 0.01 0.02 0.03 0.04 0.05 h,LM = 3.00·10−4 / w 0.62 g h,CM = 1.32·10−4 / w 0.78 g Coefficient of hygric expansion Gravimetric moisture content, wg [kg/kg] Experimental results 67 function of pore size, respectively. The criteria defined by IUPAC is followed here to classify the pore size, namely macro–pores (> 0.050 μm), meso–pores (0.002–0.050 μm), and macro–pores (< 0.002 μm) (IUPAC, 1972). It is noted that MIP does not provide information about the lower meso–pore and micro– pore ranges. As pointed out in the literature, e.g. Roels (2000), the results of the mercury intrusion porosimetry must be interpreted with caution. Since MIP is an invasion percolation method, it does not measure a true distribution of pore sizes but an ‘apparent’ pore volume distribution dependent on the connectivity of the microstructure. In other words, if large pores are only accessible by narrow throats (ink–bottle effect), they will not become filled with mercury at the expected pressure according to the Young–Laplace equation (see Eq. (3.14)). Instead, mercury will only reach these isolated large pores once the intrusion pressure has attained a value high enough to fill the finer pores (throats or passages). This may lead to an underestimation of the volume of large pores and an overestimation of the fine pore volume. (a) (b) (c) (d) Figure 4.9. MIP analysis – Cumulative porosity and percentage of porous volume occupied by each pore size: (a) brick (B); (b) lime mortar (LM); (c) cement mortar (CM); (d) lime mortar from masonry joints (LMJ). 0.01 0.1 1 10 100 0 5 10 15 20 25 30 35 40 Mesopores Pore diameter [µm] Cumulative porosity [%] Macropores Fine Mid-size Large 0 1 2 3 4 5 6 7 8 B Porous volume [%] 0.01 0.1 1 10 100 0 5 10 15 20 25 30 35 40 LM Mesopores Pore diameter [µm] Cumulative porosity [%] Macropores Fine Mid-size Large 0 5 10 15 20 Porous volume [%] 0.01 0.1 1 10 100 0 5 10 15 20 25 30 35 40 CM Mesopores Pore diameter [µm] Cumulative porosity [%] Macropores Fine Mid-size Large 0 1 2 3 4 5 6 7 8 Porous volume [%] 0.01 0.1 1 10 100 0 5 10 15 20 25 30 35 40 LMJ Mesopores Pore diameter [µm] Cumulative porosity [%] Macropores Fine Mid-size Large 0 5 10 15 20 Porous volume [%] Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 68 The percentage of porous volume is presented in Figure 4.9 for the different studied materials with the following distribution: (a) B: 80.54 % of the porous volume occupied by macro–pores, 16.63 % of the porous volume occupied by meso–pores; (b) LM: 85.03 % macro–pores, 12.03 % meso–pores; (c) CM: 64.25 % macro–pores, 31.87 % meso–pores; (d) LMJ: 76.67 % macro–pores, 18.27 % meso–pores. The remaining porous volume identified by MIP is occupied by canals or pores with diameter greater than 100 μm. Although the greatest portion of the porous system is dominated by the macro–pore volume, there is a larger number of meso–pores, as shown in Figure 4.10. The percentage of number of pores according to pore size is distributed as follows: (a) B: 99.39 % of pores between 0.004 μm and 0.050 μm and 0.61 % of pores between 0.050 μm and 100 μm; (b) LM: 99.56 % meso–pores, 0.44 % macro–pores; (c) CM: 99.79 % meso–pores, 0.21 % macro–pores; (d) LMJ: 99.62 % meso–pores, 0.38 % macro– pores. (a) (b) (c) (d) Figure 4.10. MIP analysis – Percentage of pores according to size: (a) brick (B); (b) lime mortar (LM); (c) cement mortar (CM); (d) lime mortar from masonry joints (LMJ). 0.01 0.1 1 10 100 0 4 8 12 16 20 Mesopores Mid-size LargeFine Macropores B Pore diameter [µm] Number of pores [%] 0.01 0.1 1 10 100 0 4 8 12 16 20 LM Mesopores Mid-size LargeFine Macropores Pore diameter [µm] Number of pores [%] 0.01 0.1 1 10 100 0 4 8 12 16 20 CM Mesopores Mid-size LargeFine Macropores Pore diameter [µm] Number of pores [%] 0.01 0.1 1 10 100 0 4 8 12 16 20 LMJ Mesopores Mid-size LargeFine Macropores Pore diameter [µm] Number of pores [%] Experimental results 69 (a) (b) (c) (d) Figure 4.11. MIP analysis – Differential intrusion or –d𝑉/d(log𝑑), with 𝑉 the intruded volume and 𝑑 the pore diameter, as a function of pore size: (a) brick (B); (b) lime mortar (LM); (c) cement mortar (CM); (d) lime mortar from masonry joints (LMJ). Note that the vertical axis scale is not constant. Considering the results from MIP data, the following remarks are presented as a summary of the main findings regarding the microstructure of the studied materials: a) The results for B present a broad porous distribution with most pores (75 %) between 0.02 μm and 2.00 μm diameter. Two intruded volume peaks (bi–modal distribution) are noticeable at ca. 0.10 μm and 1.00 μm, respectively (Figure 4.11a). Moreover, there is a considerable presence of large pores. Similar results may be found in the literature for ceramic brick, but they often present a more mono–modal distribution with a peak for pores around 0.80 μm to 2.00 μm in diameter (Carmeliet & Roels, 2001, 2002; Nunes et al., 2017). b) LM shows a much smaller range of pore sizes, with a maximum intruded volume at ca. 0.80 μm (Figure 4.11b). Pores larger than 1.30 μm constitute less than 10 % of the total pore volume. 0.01 0.1 1 10 100 0.00 0.02 0.04 0.06 0.08 0.10 B -dV/d(log d) [ml/g] Pore diameter [µm] Mesopores Macropores Fine Mid-size Large 0.01 0.1 1 10 100 0.00 0.04 0.08 0.12 0.16 0.20 LM -dV/d(log d) [ml/g] Pore diameter [µm] Mesopores Macropores Fine Mid-size Large 0.01 0.1 1 10 100 0.00 0.02 0.04 0.06 0.08 0.10 CM -dV/d(log d) [ml/g] Pore diameter [µm] Mesopores Macropores Fine Mid-size Large 0.01 0.1 1 10 100 0.00 0.04 0.08 0.12 0.16 0.20 LMJ -dV/d(log d) [ml/g] Pore diameter [µm] Mesopores Macropores Fine Mid-size Large Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 70 The rest of the pore system is made up by finer pores between 0.01 μm and 0.20 μm. These results are consistent with other research works, e.g. Lanas et al. (2004), Maravelaki-Kalaitzaki et al. (2005). c) The results for CM show a broad, quite uniform distribution of pores between 0.01 μm and 1.00 μm (Figure 4.11c). The presence of pores greater than 2 μm is of little significance and accounts for less than 10 % of the total pore volume. Other results in the literature support the greater presence of finer pores, with further peaks of intruded volume between 1–10 nm (Depraetere et al., 1999; Derluyn et al., 2011). These works also indicate a more marked bi– modal distribution and less presence of pores above 0.20 μm diameter. d) Finally, LMJ presents a more complex pore distribution with two main peaks, with a maximum intruded volume around 0.40 μm accounting for ca. 50 % of the pore volume, and a secondary system between 0.05–0.08 μm (Figure 4.11d). Moreover, there is a significant presence of pores bigger than 1.0 μm and up to 300 μm, which constitute around 20 % of the total pore volume. Furthermore, the distribution of these larger macro–pores is rather scattered. The different curing conditions of LM and LMJ entailed significant changes in the pore structures of the two mortars. Taking LM as the reference, LMJ presents a shift of the main peak towards smaller pore diameters together with a decrease of the intruded volume associated with this peak. Likewise, there is a shift of the secondary system towards slightly smaller diameters, but the range of finer pores shows an increase of pore intruded volume. Finally, the presence of larger pores (> 1.00 μm diameter) is more significant in LMJ. The pore size distribution of the studied materials can be associated to certain hygric properties discussed in previous sections. For instance, the larger presence of finer pores in CM (31.87 % of the porous volume is occupied by meso–pores) is directly related to its markedly hygroscopic behaviour. Similarly, LMJ revealed a slight hygroscopic behaviour and it shows a significant volume of finer pores as well (18.27 % meso–pores). On the other hand, hysteresis in the moisture storage curves, which was observed in CM and to some extent LMJ, may be linked to a particularly heterogeneous distribution of pore sizes as well as the greater presence of finer pores. In this regard, the obtained results also provide a good approximation of the observe trends. Nonetheless, it is recalled that MIP alone does not cover the whole microstructure range, so these remarks should be further complemented with detailed information about the fine meso– and micro–pore range. MIP results can be transformed into points on the (desorption) moisture retention curve with the appropriate transformations for the values of surface tension, 𝜎, and contact angle, 𝜃 (see Eq. (3.14)). Experimental results 71 In particular, for liquid water 𝜎= 0.072 N/m and 𝜃= 0°. The corresponding results are shown in Figure 4.12 together with the points obtained with static gravimetric tests and the desorption isotherms fitted using analytical models found in the literature (see Section 4.1.4). The results are consistent with moisture retention curves starting from vacuum saturation, 𝑤𝑠𝑎𝑡, which is in agreement with other studies in the literature, e.g. Roels (2000), Qiu (2003). Consequently, the moisture retention curves lay above the corresponding desorption isotherms defined gravimetrically from capillary moisture content, 𝑤𝑐𝑎𝑝, especially for lower values of suction pressure (high relative humidity). (a) (b) (c) (d) Figure 4.12. Experimental points from MIP and static gravimetric tests, and (de)sorption isotherms of the studied materials: (a) brick (B), and (b) lime mortar (LM), fitted with Künzel’s model (Künzel, 1995); (c) cement mortar (CM), and (d) lime mortar from masonry joints (LMJ), fitted with Mualem’s model (Mualem, 1976). Finally, the open porosity values obtained from MIP are presented in Table 4.5 together with the results from the evacuation method or vacuum saturation tests (see Section 4.1.1). Despite some discrepancies, mercury porosimetry tests provide results within acceptable ranges. 1 2 3 4 5 6 7 8 9 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Künzel's model Gravimetric test MIP test wsat wcap log capillary suction, log(ps) [Pa] Gravimetric moisture content [kg/kg]  =   =  1 2 3 4 5 6 7 8 9 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Künzel's model Gravimetric test MIP test wsat wcap log capillary suction, log(ps) [Pa] Gravimetric moisture content [kg/kg]  =   =  1 2 3 4 5 6 7 8 9 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Mualem's model Gravimetric test MIP test wsat wcap log capillary suction, log(ps) [Pa] Gravimetric moisture content [kg/kg]  =   =  1 2 3 4 5 6 7 8 9 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 Mualem's model Gravimetric test MIP test wsat wcap log capillary suction, log(ps) [Pa] Gravimetric moisture content [kg/kg]  =   =  Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 72 Table 4.5. Open porosity, 𝜙𝑜 [–], obtained from MIP and vacuum saturation tests. Method Material B CM LM LMJ Vacuum saturation test 0.280 0.210 0.255 0.230 Mercury Intrusion Porosimetry 0.321 0.181 0.228 0.225 Error [%] 14.64 –13.81 –10.59 –2.17 4.2 RESULTS ON MASONRY COMPOSITES The results obtained from hygric tests in multi–layered specimens are presented and discussed in this section. Two main group of experiments were performed on masonry specimens, namely capillary absorption and drying tests. 4.2.1 Capillary absorption tests The results of the capillary absorption tests for two–layer (1 interface) masonry composites, are shown in Figure 4.13 for the two studied directions, namely LMJ–to–B, labelled M1–M (Figure 4.13a), and B– to–LMJ, labelled M1–B (Figure 4.13b). The cross–sectional area of all the cylinders is the same so direct comparison of the water uptake mass is possible. Note that the tests for the configuration M1–M were concluded before the brick reached capillary saturation. (a) (b) Figure 4.13. Water absorption results for masonry specimens M1 (B+LMJ): (a) LMJ–to–B configuration, M1–M; (b) B–to– LMJ configuration, M1–B. Grey curves represent test results, and the black curve is the average. The location of the interface is estimated from the average volume of each material layer. By comparison of the two cases in Figure 4.13, it is evidenced that the uptake rate is influenced by the presence of the interface. In Figure 4.13a, the slope of the moisture inflow per square root of time falls 080 160 240 320 400 0 5 10 15 20 25 30 35 M1-M5 M1-M3, 4 Lime mortar from joint alone Brick alone Water uptake mass [g] Time [sqrt(s)] Interface 080 160 240 320 400 0 5 10 15 20 25 30 35 Water uptake mass [g] Time [sqrt(s)] Interface Experimental results 73 after a point. This point is identified as the interface and matches the expected capillary saturation for LMJ calculated as the average 𝑤𝑐𝑎𝑝,𝐿𝑀𝐽·𝑉𝑏𝑢𝑙𝑘,𝐿𝑀𝐽≅ 4.60 g. Above the interface, the uptake mass curves are less steep than expected from the absorption rate of B alone. Similarly, the location of the interface in Figure 4.13b is estimated from the capillary saturation of the brick layer calculated as the average 𝑤𝑐𝑎𝑝,𝐵·𝑉𝑏𝑢𝑙𝑘,𝐵≅ 23.50 g. However, the influence of the interface for the M1–B cases is less evident. This may be explained by several reasons, namely the smaller thickness of the mortar layer, which causes any difference to dissipate faster; and the higher suction capacity of the mortar, which counteracts the apparent retardant effect of the interface. Moreover, the further the distance of the interface with respect to the water source, the lower the interfacial effect (Vereecken et al., 2020). Dissimilar properties have been already discussed between LM and LMJ (Table 4.1). These differences were also evidenced in the water uptake tests for composite cylinders. Considering the first portion of the uptake curves in Figure 4.13a, the average water absorption coefficient for LMJ is 𝐴𝑤,𝐿𝑀𝐽= 0.076 kg/(m2·s0.5) (CoV = 7.39 %), which is about three times smaller than the coefficient obtained for LM (Table 4.3). In turn, the average water absorption of the brick counterparts (initial slope of the curves in Figure 4.13b), 𝐴𝑤,𝐵= 0.073 kg/(m2·s0.5) (CoV = 11.25 %), was slightly higher than the value obtained for brick units in the same direction (bed or Z), though still in the same range (Table 4.3). Thus, the water absorption coefficient of LMJ turned out to be similar to that of the adjacent B. From the results for M1–M (Figure 4.13a), it is possible to distinguish a global trend with upper (M1–M5) and lower (M1–M3, M1–M4) boundaries. To a greater or lesser extent, all cases revealed an absorption rate reduction associated with the presence of the interface. In general, two main factors may explain the absorption rate drop: (a) the nature and quality of the interface, namely the presence of cracks, voids, or discontinuities; and (b) the hydraulic affinity of the materials in contact, which has to do with the pore structure of each material. For instance, assuming cylindrical pores, the Young–Laplace equation (recalled here for clarity) defines the pressure at which a pore fills (or empties) according to its pore opening size: 𝑝𝑐=2𝜎 𝑟·𝑐𝑜𝑠(𝜃) (4.8) where 𝜎 [N/m] is the surface tension, 𝑟 [m] is the pore radius, and 𝜃 is the contact angle. According to Eq. (4.8), a material with finer pores will exert higher capillary pressure and thus withhold water when in contact with a material with bigger pores. Furthermore, even if the material can withdraw the water from the previous layer, its absorption rate will be limited by the behaviour of the source material. This phenomenon was defined by Wilson et al. (1995b) as ‘starvation’ of the second medium. In other words, Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 74 the effective absorption rate will be determined by the slowest layer ‘up–the–stream’. In this sense, the water uptake process in multi–layered materials may be understood as a unidirectional series circuit with specific conductive (permeability) properties for each layer and a certain hydraulic resistance associated to the interfaces between them (Brocken, 1998). The results of the capillary absorption tests for the three–layer masonry composites, labelled M2 (2 interfaces), are shown in Figure 4.14 for the two studied configurations, namely B(a)–LMJ–B(b), and the opposite, B(b)–LMJ–B(a). Direct comparison of the water uptake mass is possible since the cross– sectional area of all the cylinders is the same. The location of the interfaces in the figure is estimated from the capillary saturation levels calculated from the capillary moisture content and average volume of each material layer. Figure 4.14. Water absorption results for masonry specimens M2 (B+LMJ+B). Grey curves represent test results and black curves are the corresponding average. The location of the interfaces is estimated from the average volume of each material layer. The curves in Figure 4.14 reveal that the variability of the material response increases with the increasing number of elements involved. This is expected considering the more complex configuration of the specimens. Two main factors, namely variability of the brick layers and quality of the interfaces, give rise to the differences observed between the different specimens and the same specimens in the opposite direction. The purpose of studying the same specimens in both directions is twofold. On one hand, the absorption properties of each brick layer may be assessed individually from the first portion of the water uptake curves. This helps identifying unavoidable variability from the bricks that could otherwise obscure the results. Secondly, a two–way analysis provides information about the directional nature of the interface. Indeed, if the interface is linked to a change in pore structure, it may become apparent when the flow goes from one material to the other but not the opposite. By testing the same components in the two directions, we were able to confirm the existence of such phenomenon. 0100 200 300 400 500 600 700 0 10 20 30 40 50 60 ▼M2-4(a-to-b) M2-5(b-to-a) B(a)-to-B(b) B(b)-to-B(a) Water uptake mass [g] Time [sqrt(s)] Int.1 Int.2 Experimental results 75 As it was observed for the masonry specimens M1–M (Figure 4.13b), cases with slight, medium and high interfacial impact were detected as well for the three–layer composites (Figure 4.14). Most specimens showed hardly any absorption rate reduction between the first brick and the mortar joint. Only two cases, namely M2–4(a–to–b) and M2–5(b–to–a) (individuated in Figure 4.14), presented a considerable absorption rate drop at the first interface, most probably associated to local defects. Conversely, a more noticeable deviation was generally observed between the mortar joint and the uppermost brick. This confirms the directional behaviour of the interface and points at a microstructure change between B and LMJ. The pore quality difference between both materials is therefore assumed to be the main factor affecting the suction capacity of the following layer. In this case, the porous medium with finer pores tends to withhold water due to the higher capillary suction and thus the liquid transfer to a material with bigger pores is impeded. Considering the average behaviour of all the cases (black curves in Figure 4.14), the response of masonry specimens M2 showed no significant hydraulic resistance at the first interface (water moving from B to LMJ), whereas the second interface (water moving from LMJ to B) involved a certain flow reduction. Furthermore, the match between the averages of the two tested configurations proves that the behaviour of the composite was independent of the analysed direction. The results of the capillary absorption tests for five–layer masonry composites, labelled M4 (4 interfaces), are shown in Figure 4.15. As in the previous cases, the cross–sectional area of all the cylinders is the same, which allows the direct comparison of the water uptake mass. Moreover, the location of the interfaces in the figure is estimated from the capillary saturation levels calculated from the capillary moisture content and average volume of each material layer. Figure 4.15. Water absorption results for masonry specimens M4 (B+LMJ+B+LMJ+B). Grey curves represent test results and black curves are the corresponding average. The location of the interfaces is estimated from the average volume of each material layer. 0250 500 750 1000 1250 1500 0 15 30 45 60 75 90 Water uptake mass [g] Time [sqrt(s)] Int.1 Int.2 Int.3 Int.4 Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 82 5.1.1 Governing equations The chosen moisture transport model belongs to the so–called diffusivity approaches, which are based on Fick’s laws of diffusion: 𝜉𝜕𝜑 𝜕𝑡=∇·(𝜉𝐷𝑤 ∇𝜑)+∇·(𝛿𝑣 ∇(𝜑 𝑝𝑣,𝑠𝑎𝑡)) (5.1) where ξ=𝜕𝑤/𝜕𝜑 [kg/m3] is the moisture storage capacity, 𝑤 [kg/m3] is the moisture content, 𝜑 [–] is the relative humidity, 𝑡 [s] is the time, 𝐷𝑤 [m2/s] is the liquid water diffusivity, 𝛿𝑣 [kg/(m·s·Pa)] is the water vapour permeability, 𝑝𝑣,𝑠𝑎𝑡 [Pa] is the saturation vapour pressure. In turn, the water vapour permeability is defined as: 𝛿𝑣=𝛿𝑎 𝜇 (5.2) where δa [kg/m3] is the water vapour permeability of still air, and 𝜇 [–] is the vapour diffusion resistance factor of the porous material. In Eq. (2.1), the first term on the right–hand side represents liquid water transport, whereas the second term stands for water vapour diffusion. It must be noted that the moisture content is usually expressed either as mass of adsorbed water per volume of dry material, 𝑤 [kg/m3], as mass of adsorbed water per mass of dry material, 𝑤𝑔 [kg/kg], known as gravimetric moisture content, or as volume of adsorbed water per volume of dry material, 𝑤𝑉 [m3/m3], known as volumetric moisture content. The three expressions are related through: 𝑤=𝑤𝑔·𝜌𝑏𝑢𝑙𝑘=𝑤𝑉·𝜌𝑤 (5.3) where 𝜌𝑏𝑢𝑙𝑘 [kg/m3] is the bulk density of the porous material, and 𝜌𝑤 [kg/m3] is the density of liquid water. For the sake of simplicity, isothermal conditions are assumed, thus cancelling additional thermal influences that could affect temperature–dependent parameters, such as relative humidity, water vapour permeability, etc. Different analytical formulations may be found in the literature to account for the liquid water diffusivity. For the present study, a modified version of the exponential expression proposed by Künzel (1995) is used: 𝐷𝑤=𝛾·(𝐴𝑤 𝑤𝑐𝑎𝑝)2·103·( 𝑤 𝑤𝑐𝑎𝑝 −1) (5.4) Numerical simulations I: Moisture transport 83 where 𝛾 [–] is a diffusivity factor depending on the material and transport process (wetting/drying), 𝐴𝑤 [kg/(m2·s0.5)] is the capillary absorption coefficient, and 𝑤𝑐𝑎𝑝 [kg/m3] is the capillary moisture content. The original formulation by Künzel uses a fixed value 𝛾= 3.80. In this work, 𝛾 is left as an adjustable parameter in order to accommodate a variety of material transport processes. The main advantage of the chosen expression comes from the fact that it is defined by means of reasonably simple measurable properties, namely the capillary absorption coefficient, and it has only one adjustable parameter, which nonetheless guarantees a great deal of flexibility. 5.1.2 Interface modelling The simulation of moisture transport in multi–layered materials such as masonry requires specific considerations due to the combination of media with different hygric properties and the presence of interfacial zones between consecutive layers (Brocken, 1998; Derluyn et al., 2011; Janssen et al., 2012; X. Zhou et al., 2020). In the context of the present studies, the interfacial zone between bricks and mortar is treated macroscopically, thus a phenomenological interface with zero thickness is assumed. Considering the combination of dissimilar materials, due to the discontinuity between their moisture storage properties, a formulation based on a continuous potential, such as capillary pressure, vapour pressure, or (as the case here) relative humidity is necessary. Then, the type of interface contact must be considered. If the transition from one layer to the other has no impact on moisture transport, the interface is assumed to have perfect hydraulic contact. However, in most multi–layered materials, some retardation of the moisture flux across the interface is observed, which reveals the existence of an imperfect hydraulic contact. This phenomenon can be attributed to the discontinuity between the pore structures of the materials (natural contact), the existence of an air gap between the adjacent layers, or a combination of both cases (Brocken, 1998). In order to account for these phenomena, Brocken (1998) proposed the introduction of a parameter to describe the interface permeability, 𝐾𝐼𝐹 [s/m], or alternatively an interface resistance, 𝑅𝐼𝐹 [m/s]. It is noted that the interface (either perfect or imperfect contact) has zero thickness and no hygroscopic capacity. If a constant interface resistance is assumed, the water flow across the interface, 𝑔𝐼𝐹 [kg/(m2· s)], can be described by: 𝑔𝐼𝐹=𝐾𝐼𝐹𝜕𝑝𝑐 𝜕𝑥 =1 1/𝐾𝐼𝐹𝜕𝑝𝑐 𝜕𝑥 =∆𝑝𝑐 1 𝐾𝐼𝐹·∆𝑥=∆𝑝𝑐 𝑅𝐼𝐹 (5.5) where 𝑝𝑐 [Pa] is the capillary pressure. Therefore, the imperfect hydraulic contact translates into a drop in capillary pressure across the interface. Since the presented model uses relative humidity as driving Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 84 potential, a change of variable becomes necessary. It is recalled that Kelvin equation can be used to link relative humidity with capillary pressure, such as: 𝑝𝑐=𝜌𝑤·𝑅𝑣·𝑇·𝑙𝑛𝜑 (5.6) where 𝜌𝑤 [kg/m3] is the density of water, 𝑅𝑣 [J/(kg·K)] is the universal gas constant for water vapour, and 𝑇 [K] is the absolute temperature. Assuming the applicability of Kelvin’s equation, the water flow across the interface can be redefined with respect to the relative humidity: 𝑔𝐼𝐹=𝐾𝐼𝐹𝜕𝑝𝑐 𝜕𝜑𝜕𝜑 𝜕𝑥=𝐾𝐼𝐹𝜌𝑤·𝑅𝑣·𝑇 𝜑𝜕𝜑 𝜕𝑥=𝜌𝑤·𝑅𝑣·𝑇 𝜑∆𝜑 1 𝐾𝐼𝐹·∆𝑥=𝜌𝑤·𝑅𝑣·𝑇 𝜑∆𝜑 𝑅𝐼𝐹 (5.7) 5.1.3 Initial and boundary conditions The model is completed by introducing the initial conditions and the boundary conditions (BC). In particular, the boundary conditions can be of Dirichlet (also called first) or Neumann (second) type. The Dirichlet BC indicates: 𝜑=𝜑 (5.8) where 𝜑 [–] is the prescribed relative humidity at the boundary. Conversely, the Neumann BC is defined as a flux derived from a vapour pressure difference: 𝑔=ℎ𝑚(𝑝𝑣−𝑝𝑣,𝑒𝑥𝑡) (5.9) where 𝑔 [kg/(m2·s)] is the convective moisture flux, ℎ𝑚 [s/m] is the convective mass transfer coefficient, and 𝑝𝑣,𝑒𝑥𝑡 [Pa] and 𝑝𝑣 [Pa] are the partial vapour pressures defined for the environment and at the model boundary, respectively. Partial vapour pressure and relative humidity are related by: 𝜑= 𝑝𝑣 𝑝𝑣,𝑠𝑎𝑡(𝑇) (5.10) where the saturation vapour pressure, 𝑝𝑣,𝑠𝑎𝑡 [Pa], may be described empirically as a non–linear function of temperature, 𝑇 [K], as defined in Chapter 3: 𝑝𝑣,𝑠𝑎𝑡=610.7[Pa]·107.5·(𝑇−273.15 𝑇−35.85) (5.11) Numerical simulations I: Moisture transport 85 5.2 NUMERICAL SIMULATIONS AND MODEL VALIDATION The moisture transport model described in Section 5.1 is applied in the following to simulate the hygric behaviour of single materials (brick and mortar) and multi–layered (masonry) specimens. The material properties presented in Chapter 4 are used as numerical input parameters. For the sake of clarity, a summary of the material properties of interest is given in Table 2.2. Similarly, the moisture storage curves used for each material are presented in Figure 4.2. Table 5.1. Summary of material properties used for moisture transport simulations. Material Property Symbol Value Units Source B Bulk density 𝜌𝑏𝑢𝑙𝑘 1900 kg/m3 Experimental Open porosity 𝜙𝑜 0.280 – Experimental Capillary moisture content 𝑤𝑐𝑎𝑝 240 kg/m3 Experimental Fitting parameter for sorption isotherm 𝜓 1.0070 – Experimental Water absorption coefficient (X–direction) 𝐴𝑤,𝑋 0.104 kg/(m2·s0.5) Experimental Water absorption coefficient (Y–direction) 𝐴𝑤,𝑌 0.089 kg/(m2·s0.5) Experimental Water absorption coefficient (Z–direction) 𝐴𝑤,𝑍 0.061.(1) kg/(m2·s0.5) Experimental Water vapour resistance factor 𝜇 34.14 – Experimental CM Bulk density 𝜌𝑏𝑢𝑙𝑘 2000 kg/m3 Experimental Open porosity 𝜙𝑜 0.210 – Experimental Capillary moisture content 𝑤𝑐𝑎𝑝 180 kg/m3 Experimental Fitting parameter for adsorption isotherm 𝑎𝑎𝑑𝑠 1.43E–6 1/Pa Experimental 𝑚𝑎𝑑𝑠 0.285 – Experimental Fitting parameter for desorption isotherm 𝑎𝑑𝑒𝑠 2.51E–5 1/Pa Experimental 𝑚𝑑𝑒𝑠 0.213 – Experimental Water absorption coefficient (isotropic) 𝐴𝑤,𝐼𝑆𝑂 0.060 kg/(m2·s0.5) Experimental Water vapour resistance factor 𝜇 25.00 – Prangnell (1971) LM Bulk density 𝜌𝑏𝑢𝑙𝑘 1990 kg/m3 Experimental Open porosity 𝜙𝑜 0.255 – Experimental Capillary moisture content 𝑤𝑐𝑎𝑝 225 kg/m3 Experimental Fitting parameter for sorption isotherm 𝜓 1.0066 – Experimental Water absorption coefficient (isotropic) 𝐴𝑤,𝐼𝑆𝑂 0.235 kg/(m2·s0.5) Experimental Water vapour resistance factor 𝜇 15.00 – Prangnell (1971) .(1) For multi–layered cases, 𝐴𝑤,𝐵−𝑍= 0.075 kg/(m2·s0.5) Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 86 Table 5.1 (Continued). Summary of material properties used for moisture transport simulations. Material Property Symbol Value Unit Source LMJ Bulk density 𝜌𝑏𝑢𝑙𝑘 2060 kg/m3 Experimental Open porosity 𝜙𝑜 0.230 – Experimental Capillary moisture content 𝑤𝑐𝑎𝑝 190 kg/m3 Experimental Fitting parameter for adsorption isotherm 𝑎𝑎𝑑𝑠 3.80E–6 1/Pa Experimental 𝑚𝑎𝑑𝑠 0.352 – Experimental Fitting parameter for desorption isotherm 𝑎𝑑𝑒𝑠 1.19E–5 1/Pa Experimental 𝑚𝑑𝑒𝑠 0.235 – Experimental Water absorption coefficient (isotropic) 𝐴𝑤,𝐼𝑆𝑂 0.080 kg/(m2·s0.5) Experimental Water vapour resistance factor 𝜇 15.00 – Prangnell (1971) The simulations presented herein are based on finite element analysis and were performed using COMSOL Multiphysics (COMSOL, 2021). Since the analysed cases can be reduced to a 1–D phenomenon, linear elements with quadratic shape functions were used for the discretization of the model. Preliminary sensitivity analyses were performed to determine the optimal mesh size and, as a result, the average element size for all the models was set to 1 mm. The primary variable, i.e. the relative humidity 𝜑, was interpolated based on standard Lagrangian shape functions. Moreover, the backward finite difference method was used for automatic time discretization during the analysis. Finally, an iterative procedure based on the Newton–Raphson method was applied to solve the non–linear differential equations. The imposed boundary conditions are summarized as follows: a) Water absorption simulations; 𝜑= 1 was defined at the base to represent liquid water, whereas null flux condition 𝑔= 0 kg/(m2·s) was imposed at the top face. The boundary condition at the base was introduced progressively using a smoothed step function. b) Drying simulations; the bottom node was insulated whereas a convective flux was imposed at the top face, such as 𝑔=ℎ𝑚(𝑝𝑣−𝑝𝑣,𝑒𝑥𝑡), with ℎ𝑚= 6.05E–8 s/m. The partial vapour pressure of the environment was defined assuming 𝜗= 23 °C and 𝜑= 0.55. Note that the value for the convective mass transfer coefficient was calculated from the drying tests performed on single materials, as explained in Section 5.2.2. Additionally, the initial conditions were taken as: a) Water absorption simulations; 𝜑(𝑡=0)= 0, i.e. dry conditions. For the multi–layered configurations, 𝜑= 0 is incompatible with the definition of the moisture transfer coefficient Numerical simulations I: Moisture transport 87 applied at the interfaces, see Eq. (5.7). Hence, for such cases a small value was assumed, i.e. 𝜑(𝑡=0)= 0.01, which can still be considered equivalent to the dry state. b) Drying simulations; 𝜑(𝑡=0)= 1, i.e. capillary saturation. Finally, a qualitative curve–fitting process was used to estimate the necessary hygric properties, namely the liquid water diffusivity for drying and the interface hydraulic resistance. Thus, the average trend of the measured data was used for comparison against the simulated curves to calibrate the corresponding parameters until an optimized solution was found. It must be noted that, in this context, the accuracy of the model is considered based on the predictions of water uptake mass and moisture mass loss in capillary absorption and drying tests, respectively. Therefore, the global response of the system is used as a reference for validation. It is recalled, however, that a full validation would require the additional study of moisture profiles inside the specimens, which is out of the scope of the present work. (a) (b) (c) (d) Figure 5.1. Moisture storage curves used for simulations: (a) brick (B), and (b) lime mortar (LM), fitted with Künzel’s model (Künzel, 1995); (c) cement mortar (CM), and (d) lime mortar from masonry joints (LMJ), fitted with Mualem’s model (Mualem, 1976). 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Exp. Adsorption Exp. Desorption Künzel's model Gravimetric moisture content [kg/kg] Relative humidity,  [-] 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Relative humidity,  [-] Gravimetric moisture content [kg/kg] Exp. Adsorption Exp. Desorption Künzel's model 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Relative humidity,  [-] Gravimetric moisture content [kg/kg] Exp. Adsorption Exp. Desorption Mualem's model - Adsorption Mualem's model - Desorption 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Relative humidity,  [-] Gravimetric moisture content [kg/kg] Exp. Adsorption Exp. Desorption Mualem's model - Adsorption Mualem's model - Desorption Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 88 In parallel to the development of the following sections, a series of parametric studies was carried out to understand the influence of each model parameter in the hygric response. These parametric studies are collected in a dedicated appendix, where the corresponding discussion and conclusions are also presented. The reader is referred to Appendix 3 for further details. 5.2.1 Water absorption in single materials The results for the capillary absorption tests on single materials, namely B, CM and LM, are shown in Figure 5.2 together with the simulated curves. From the comparison between experimental and numerical results, it is clear that the proposed moisture transport model can capture the observed behaviour with great accuracy. For all the absorption cases, the diffusivity factor introduced in Eq. (5.4) was defined using the original value of 𝛾 proposed by Künzel (1995), i.e. 𝛾𝑎𝑑𝑠= 3.80. Two other parameters were needed to determine the water uptake response of the model, namely (a) the capillary absorption coefficient, 𝐴𝑤, and (b) the capillary moisture content, 𝑤𝑐𝑎𝑝. Note that both parameters were directly obtained from capillary absorption tests. As demonstrated by the numerical simulations, the capillary absorption coefficient is related to the water absorption rate: the higher 𝐴𝑤, the steeper the mass gain with respect to the square root of time. In other words, higher 𝐴𝑤 values entail a faster absorption process, as expected. This relation is clearly evidenced by the studied materials since 𝐴𝑤,𝐿𝑀−𝐼𝑆𝑂 > 𝐴𝑤,𝐵−𝑋 > 𝐴𝑤,𝐵−𝑌 > 𝐴𝑤,𝐵−𝑍 ≈ 𝐴𝑤,𝐶𝑀−𝐼𝑆𝑂. On the other hand, 𝑤𝑐𝑎𝑝 determines the equilibrium plateau attained at the end of the absorption process, which can be verified from the volume of the specimen, 𝑉, such as 𝑉 [m3] × 𝑤𝑐𝑎𝑝 [kg/m3]. 5.2.2 Drying in single materials The results of the drying tests on single materials, namely B, CM and LM, are shown in Figure 5.3 together with the simulated curves. Considering the consistency between experimental and numerical results, it can be concluded that the model is able to reproduce the drying behaviour with excellent accuracy. For these cases, an adjustment of the liquid water diffusivity function (Eq. (5.4)) was needed to match the experimental results. In particular, the diffusivity factor 𝛾 was indirectly estimated to calibrate the numerical response against the experimental data. The updated values of the diffusivity factor 𝛾 employed for drying cases, i.e. 𝛾𝑑𝑒𝑠, are collected in Table 4.4. Numerical simulations I: Moisture transport 89 (a) (b) (c) (d) (e) Figure 5.2. Capillary absorption results for single materials: (a) B, X or extrusion direction; (b) B, Y or stretcher direction; (c) B, Z or bed direction; (d) CM, isotropic; (e) LM, isotropic. 040 80 120 160 200 240 0 4 8 12 16 20 Time [sqrt(s)] Water uptake mass [g] Exp.X Exp.X Avg. Sim.X 040 80 120 160 200 240 0 4 8 12 16 20 Exp.Y Exp.Y Avg. Sim.Y Time [sqrt(s)] Water uptake mass [g] 040 80 120 160 200 240 0 4 8 12 16 20 Time [sqrt(s)] Exp.Z Exp.Z Avg. Sim.Z Water uptake mass [g] 040 80 120 160 200 240 0 4 8 12 16 20 Exp.CM Exp.CM Avg. Sim.CM Time [sqrt(s)] Water uptake mass [g] 040 80 120 160 200 240 0 4 8 12 16 20 Exp.LM Exp.LM Avg. Sim.LM Time [sqrt(s)] Water uptake mass [g] Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 90 (a) (b) (c) (d) (e) Figure 5.3. Drying results for single materials: (a) B, X or extrusion direction; (b) B, Y or stretcher direction; (c) B, Z or bed direction; (d) CM, isotropic; (e) LM, isotropic. 0100 200 300 400 500 0 5 10 15 20 Time [h] Mass loss [g] Exp.X Exp.X Avg. Sim.X  ads =  des 0100 200 300 400 500 0 5 10 15 20 Time [h]  des =   ads  ads Mass loss [g] Exp.Y Exp.Y Avg. Sim.Y 0100 200 300 400 500 0 5 10 15 20 Time [h]  ads Mass loss [g] Exp.Z Exp.Z Avg. Sim.Z  des =   ads 0100 200 300 400 500 0 5 10 15 20 Time [h]  ads Exp.CM Exp.CM Avg. Sim.CM Mass loss [g]  des = 0.27  ads 0100 200 300 400 500 0 5 10 15 20  ads Exp.LM Exp.LM Avg. Sim.LM Mass loss [g] Time [h]  des = 0.18  ads Numerical simulations I: Moisture transport 91 The adjustment of the original liquid diffusivity function is in agreement with the studies by Scheffler (2008), who discussed the existence of hysteretic transport functions for material models based on the diffusivity approach (Fick’s law). Likewise, Krus (1996) explained this phenomenon by the different velocities for wetting and drying processes. In other words, since drying and wetting occur at different rates, their corresponding liquid transport coefficients can differ. Therefore, two diffusivity functions are necessary, namely one for adsorption, 𝐷𝑤,𝑎𝑑𝑠, and one for desorption, 𝐷𝑤,𝑑𝑒𝑠. However, no analytical expression is available in the literature specifically for drying. As an indication, Künzel (1995) showed that for certain porous stones, the liquid diffusivity for desorption at capillary saturation could be approximately 3–5 times (finely porous stone) to one order of magnitude (coarse porous stone) lower than the diffusivity for adsorption at 𝑤𝑐𝑎𝑝. Other studies demonstrated that the diffusivity for desorption can be adjusted by means of numerical simulations based on drying experiments (Scheffler, 2008). For instance, Krus & Holm (1999) proposed an analytical expression for the liquid water diffusivity as a function of moisture content. Their formulation can be directly applied to absorption cases, whereas for drying, a curve fitting process must be used to determine the diffusivity. For moisture conditions below 50 % RH, these authors proposed a fixed value 𝐷𝑤= 2.0E+10 m2/s, applicable to both transport processes. For higher moisture contents, however, drying would require a lower liquid transport coefficient. In particular, for their studied set of building porous materials, the relation 𝐷𝑤,𝑎𝑏𝑠/𝐷𝑤,𝑑𝑒𝑠 at capillary saturation ranged between 1.5 for fired–clay brick, 2.0–3.0 for natural stones and up to 5.0 for lime silica brick. In the present study, the relation 𝐷𝑤,𝑎𝑏𝑠/𝐷𝑤,𝑑𝑒𝑠 at capillary saturation reached 3.70 for CM and 5.67 for LM (Table 4.4), which is consistent with the literature and points out a significantly slower drying process compared to wetting. In fact, the drying simulation performed for both mortars using the adsorption diffusivity, i.e. 𝛾𝑎𝑑𝑠= 3.80, resulted into a much quicker mass loss with respect to the experimental evidence (see Figure 5.3d and Figure 5.3e). Interestingly, the behaviour of the extruded fired–clay brick followed a different trend. The drying simulation for the extrusion direction, BX, did not require any update to fit the experimental results, that is 𝛾𝑑𝑒𝑠=𝛾𝑎𝑑𝑠 (Figure 5.3a). For the other two directions, however, the numerical results obtained using 𝛾𝑎𝑑𝑠 resulted into lower mass loss rates when compared with the experimental data (Figure 5.3b and Figure 5.3c). Thus, 𝐷𝑤,𝑎𝑏𝑠/𝐷𝑤,𝑑𝑒𝑠< 1 for the stretcher and the bed directions of the brick, BY and BZ, respectively. In other words, higher diffusivity values were needed to match the tests results. This behaviour seems contrary to most trends described in the literature for porous building materials. However, a possible explanation can be found in the orthotropic nature of extruded bricks due to their manufacturing process. In fact, when analysing the diffusivity values with respect to the extrusion direction, significant relations come to light (see Table 4.4). Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 98 5.3.1 Numerical implementation Inspired by the work of Zhang et al. (2015), a hysteresis index 𝑊𝑜𝑟𝐷 (‘wetting or drying’) is introduced and assigned to each element during the numerical simulations. This auxiliary index ranges between 0 and 1, and indicates the current state of each node, namely drying (𝑊𝑜𝑟𝐷= 0) or wetting (𝑊𝑜𝑟𝐷= 1). The initial value of 𝑊𝑜𝑟𝐷 must be defined manually at the beginning of the analysis according to the initial step of the process. Then, the index is automatically updated by examining the sign of the relative humidity difference between the current time step and the previous one, ∆𝜑(𝑖). Assuming initial wetting, if ∆𝜑(𝑖) is positive, then the material is indeed wetting, so 𝑊𝑜𝑟𝐷 is kept equal to 1. Conversely, if ∆𝜑(𝑖) becomes negative, then the material has changed to drying and 𝑊𝑜𝑟𝐷 transitions towards 0. Thus, the switching expression is defined as: 𝜕𝑊𝑜𝑟𝐷 𝜕𝑡 =𝑖𝑓(𝜕𝜑 𝜕𝑡>0,𝑉𝑝𝑜𝑠,𝑉𝑛𝑒𝑔)+𝑆𝑡𝑒𝑝𝑁𝑒𝑔(𝑊𝑜𝑟𝐷)+𝑆𝑡𝑒𝑝𝑃𝑜𝑠(𝑊𝑜𝑟𝐷) (5.14) where 𝑉𝑝𝑜𝑠=−𝑉𝑛𝑒𝑔 are control parameters defining the transition between the target values, and 𝑆𝑡𝑒𝑝𝑁𝑒𝑔 and 𝑆𝑡𝑒𝑝𝑃𝑜𝑠 are two correction functions. In this work, 𝑉𝑝𝑜𝑠= 1.0E–5 and 𝑉𝑛𝑒𝑔= –1.0E–5. It is worth noting that 𝑉𝑝𝑜𝑠 and 𝑉𝑛𝑒𝑔 can be tuned and set to higher values, which will make the transition between the two target functions more abrupt. However, a sharper transition may cause convergence problems. Moreover, 𝑆𝑡𝑒𝑝𝑁𝑒𝑔 and 𝑆𝑡𝑒𝑝𝑃𝑜𝑠 are defined as smooth step functions which become active when 𝑊𝑜𝑟𝐷 goes below 0 or above 1, and so redirect the index towards values within the pre–established range. As an additional safeguard, the hysteresis index may be replaced by an expression such as “max(min(𝑊𝑜𝑟𝐷,1)0)”. The max() and min() operators return the maximum and minimum value of the two arguments so they ensure that the hysteresis index cannot become smaller than 0 or larger than 1. Once the hysteresis parameters have been defined, the resulting index is introduced in the expression for the liquid water diffusivity function: 𝐷𝑤(𝜑)=𝑊𝑜𝑟𝐷·𝐷𝑤,𝑎𝑑𝑠(𝜑)+(1−𝑊𝑜𝑟𝐷)·𝐷𝑤,𝑑𝑒𝑠(𝜑) (5.15) The implementation of this hysteresis index ensures that the proposed moisture transport model can adjust automatically to any arbitrary wetting/drying cycles. Hysteresis was observed as well in the sorption isotherms of CM and LMJ, both materials with marked hygroscopic behaviour. In order to capture the moisture storage hysteresis, a procedure analogous to the one applied for the liquid diffusivity function can be used. More specifically, an additional control parameter 𝐴𝑜𝑟𝐷 (‘adsorption or desorption’) must be defined as in Eq. (5.14) and applied to the moisture storage function of the material as in Eq. (5.15). Numerical simulations I: Moisture transport 99 5.3.2 Benchmark models A simple benchmark is proposed herein in order to validate the hysteresis model proposed for the liquid transport coefficient 𝐷𝑤. The model is based on data collected from capillary absorption and drying tests for cement mortar cubes (see Sections 5.2.1 and 5.2.2). This material is chosen since it presented significant differences between wetting and drying responses. Hence, 1–D numerical models are prepared to simulate the following scenarios: (a) water absorption with subsequent drying; (b) drying followed by water absorption. Case study I: Absorption followed by drying An initially dry mortar cube is subjected to free water uptake; the lower end is in contact with water (Dirichlet BC, 𝜑= 1) whereas null flux condition (Neumann BC, 𝑔= 0 kg/(m2·𝑠)) is imposed on the opposite side. Once capillary saturation has been attained, the boundary conditions change so that the mortar undergoes drying; null flux condition is defined at the bottom, whereas a convective flux condition given by 𝑔=ℎ𝑚(𝑝𝑣−𝑝𝑣,𝑒𝑥𝑡) is imposed at the top. The absorption conditions are maintained for the first 24 hours, whereas drying follows in the period 24–500 hours. Both processes take place under controlled environmental conditions, namely 23 ℃ and 55 % RH. The results obtained for the first case study, i.e. wetting followed by drying, are presented in Figure 5.6. Three different scenarios are distinguished. First, the two processes are modelled separately, each one with the corresponding liquid diffusivity function, namely 𝐷𝑤,𝑎𝑑𝑠 for water absorption and 𝐷𝑤,𝑑𝑒𝑠 for drying. Subsequently, both processes are modelled continuously using a single diffusivity function, in this case 𝐷𝑤,𝑎𝑑𝑠, which corresponds to the initial wetting phase. The new drying curve shows a clear deviation from the results obtained with the independent drying simulation. In particular, the moisture mass loss is faster for the model using 𝐷𝑤,𝑎𝑑𝑠 to describe the whole process. This is expected since 𝐷𝑤,𝑎𝑑𝑠> 𝐷𝑤,𝑑𝑒𝑠 for the case of cement mortar (see Section 5.2.2). Finally, the hysteresis model is implemented so that the diffusivity function is automatically updated according to the type of process. As in the previous case, water absorption and drying are modelled continuously. Moreover, several calculations are performed using different values for 𝑉𝑝𝑜𝑠=−𝑉𝑛𝑒𝑔. In general, the obtained results reproduce the target behaviour accurately. In particular, the moisture mass curve obtained with the hysteresis model 𝑉𝑝𝑜𝑠= −𝑉𝑛𝑒𝑔= 5.0E–5 perfectly matches the curves calculated with the independent analyses. Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 100 Figure 5.6. Moisture transport simulations to validate the hysteresis model. Case study I: wetting followed by drying. Note that two different scales are used for the horizontal axis. Case study II: Drying followed by absorption An initially capillary–saturated mortar cube is subjected to drying from a single face; a convective flux condition 𝑔=ℎ𝑚(𝑝𝑣−𝑝𝑣,𝑒𝑥𝑡) is defined for the exposed surface, whereas null flux condition is imposed on the opposite side. After 500 hours, the boundary conditions change to represent capillary absorption; the bottom is placed in direct contact with water (𝜑= 1), and null flux is imposed at the top. The water absorption conditions are maintained for 24 hours. Both processes occur under controlled environmental conditions, namely 23 ℃ and 55 % RH. The results obtained for the second case study, i.e. drying followed by wetting, are presented in Figure 5.7. Once more, three different scenarios are identified. First, the two processes are modelled separately, each one with the corresponding liquid diffusivity function, namely 𝐷𝑤,𝑑𝑒𝑠 for drying and 𝐷𝑤,𝑎𝑑𝑠 for wetting. Subsequently, both processes are modelled continuously using a single diffusivity function, in this case 𝐷𝑤,𝑑𝑒𝑠 since it corresponds to the initial drying. The moisture mass curve obtained with the new simulation shows a clear deviation with respect to the water absorption results calculated independently. In particular, the moisture mass gain is slower for the model using 𝐷𝑤,𝑑𝑒𝑠 to describe the whole process. This is due to the lower diffusivity for desorption, i.e. 𝐷𝑤,𝑑𝑒𝑠<𝐷𝑤,𝑎𝑑𝑠, observed for cement mortar (see Section 5.2.2). Finally, the hysteresis model is implemented so that the diffusivity function is automatically updated according to the type of process. As in the previous case, drying and water absorption are modelled continuously. Moreover, different values of 𝑉𝑝𝑜𝑠=−𝑉𝑛𝑒𝑔 are used for the calculations. It is noted that for the same 𝑉𝑝𝑜𝑠=−𝑉𝑛𝑒𝑔 values, the hysteresis model gives less accurate results than for the previous case study, i.e. wetting followed by drying. This fact is not surprising since water 0 1 2 3 4 0 3 6 9 12 15 24 100 200 300 400 500 Wetting (independent) Dw,ads Moisture mass [g] Time [sqrt(h)] Drying (independent) Dw,des Wetting + Drying Dw,ads Wetting + Drying Hysteresis Vpos = 1E-5 Wetting + Drying Hysteresis Vpos = 2E-5 Wetting + Drying Hysteresis Vpos = 5E-5 Time [h] Numerical simulations I: Moisture transport 101 absorption is a much faster process, and a quicker transition would be necessary. Thus, higher values of the controlling parameters 𝑉𝑝𝑜𝑠=−𝑉𝑛𝑒𝑔 should be used: higher values entail a quicker transition between both diffusivity functions and consequently produce more accurate solutions. However, as it was mentioned, sharper transitions between the diffusivity functions may induce numerical instabilities. Indeed, a model with 𝑉𝑝𝑜𝑠=−𝑉𝑛𝑒𝑔= 1.0E–4 was tried, but it suffered from convergence problems. Nevertheless, it is clear that the implementation of hysteresis implies a considerable improvement with respect to the scenario where only one diffusivity function is used. Figure 5.7. Moisture transport simulations to validate the hysteresis model. Case study II: drying followed by wetting. Note that two different scales are used for the horizontal axis. 5.4 MODELLING STRATEGIES From a mechanical analysis point of view, different computational strategies have been reported in the literature to deal with the study of masonry and masonry structures. Figure 3.4 summarizes the most common approaches. In terms of complexity, there are two major types of modelling strategies, namely micro– and macro–modelling (Lourenço, 1996). Several phenomenological models accounting for the microstructure of the material represent a further development within the micro–modelling approach (Petracca et al., 2017). The micro–modelling strategies account for the mechanical behaviour of masonry by means of non–linear interface elements, continuum elements with non–linear behaviour or a combination of both. In turn, macro–modelling strategies assume a homogenized continuum material usually described by non–linear constitutive laws. Depending on the masonry texture or the level of complexity of the study, macro–models may assume isotropic or orthotropic continua. Although the focus of these structural mechanics studies is usually different, several lessons may be learned from a cross–disciplinary approach. In this sense, certain parallelisms may be drawn between the mechanical and the moisture transport (possibly extendable to hygrothermal) modelling strategies. 0100 200 300 400 500 0 3 6 9 12 15 22.5 22.6 22.7 22.8 22.9 23.0 Moisture mass [g] Time [h] Drying (independent) Dw,des Wetting (independent) Dw,ads Drying + Wetting Dw,des Drying + Wetting Hysteresis Vpos = 1E-5 Drying + Wetting Hysteresis Vpos = 2E-5 Drying + Wetting Hysteresis Vpos = 5E-5 Time [sqrt(h)] Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 102 Moreover, couplings between the different fields may be created as well. For this type of multi–physics analyses, the link between the hygric and mechanical fields is usually established through a one–way simple coupling process. This implies that the moisture distribution is calculated first and the obtained hygric strains are used as initial input for the mechanical analysis by means of the total strain decomposition principle (Ramézani & Jeong, 2011). The influence of moisture content on mechanical properties may be considered as well (Carmeliet, 2015). Figure 5.8. Modelling strategies for masonry. Adapted from D’Altri et al. (2018), following Lourenço (1996), and Petracca et al. (2017). In Sections 5.2.3 and 5.2.4, an application of the detailed micro–modelling approach (Figure 3.4a) was presented and validated for water absorption and drying processes. The following subsections describe how the moisture transport simulations can be extended to other modelling strategies commonly used for mechanical studies of masonry. The same experimental data previously presented were used to calibrate the necessary model parameters and evaluate the accuracy of the simulations. In particular, the capillary absorption tests performed on masonry specimens M2 and M4 were employed as case studies. Thus, the corresponding moisture transport analyses were performed assuming the other strategies presented in Figure 3.4, namely continuous micro–modelling approach (Figure 3.4b), discrete micro– modelling approach (Figure 3.4c), and macro–modelling approach (Figure 3.4d). Additionally, a comparison between the different strategies is presented in terms of accuracy, level of complexity, flexibility, requirements, and limitations. In general, an iterative curve–fitting procedure was used to calibrate the numerical parameters and obtain a good correlation between simulated and experimental results. In particular, the models that explicitly account for the interfaces (detailed micro– and discrete micro–) were calibrated by tuning the Numerical simulations I: Moisture transport 103 interface resistance 𝑅𝐼𝐹 whereas the models without interfaces (continuous micro– and macro–) were calculated using the diffusivity factor 𝛾 as fitting parameter. Moreover, an averaging procedure was used to calculate the equivalent properties needed for the discrete micro– and the macro–modelling approaches, which rely on homogenization. This concept is based on the volume fraction of each material with respect to the total volume, in the original configuration. Therefore, an equivalent property 𝑋𝐸𝑄 is calculated as: 𝑋𝐸𝑄=∑ 𝑋𝑖𝑉𝑖 𝑉𝑇𝑜𝑡𝑎𝑙 𝑛 𝑖 (5.16) where 𝑋 represents the original value of the studied parameter, 𝑉 is the volume, and 𝑖 represents the material layer in a 𝑛–layered composite. It must be recalled that the constituent materials, B and LMJ, have been defined in the present work by two different moisture storage models (Figure 4.2). In the current section, the distinction between the sorption isotherms was kept for the models with explicit consideration of the two materials (detailed micro– and continuous micro–). However, for the models with an equivalent continuum (discrete micro– and macro–), a simplification was employed, and the sorption isotherm of the brick alone (Künzel’s model) was chosen to be representative of the equivalent behaviour. It is noted that this generalization is applicable to capillary–active materials and absorption processes, in which the porous medium is exposed to high relative humidity boundary conditions. In such circumstances, mass transport is governed mainly by liquid water movement (Zhang & Scherer, 2018). In other words, the absorption process involves the uppermost portion of the moisture storage curve. However, the same generalization may prove faulty in drying simulations with hygroscopic materials, such as LMJ. For such cases, the equivalent behaviour could be captured more accurately through a Mualem’s type formulation (see Eq. (4.3)). 5.4.1 Continuous micro–modelling The moisture transport equation and boundary conditions presented in Section 5.1 were applied to simulate the capillary absorption tests in M2 and M4 specimens by means of a continuous micro– modelling strategy (Figure 3.4b). For the continuous micro–modelling approach, brick and mortar were modelled with their original properties, that is, as obtained experimentally or with the corresponding values taken from the literature (Table 2.2). In order to account for the interfacial impact on moisture flux, the diffusivity factor 𝛾 associated to the mortar joints was tuned until a good agreement with the experimental data was found. Hence, the necessary parameters and updated material properties are presented in Table 5.3 and the corresponding results are shown in Figure 5.9a and Figure 5.9b. Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 104 Note that when the original 𝛾𝑎𝑑𝑠,𝐿𝑀𝐽= 3.80 is used, the problem is equivalent to the detailed micro– model with perfect contact (no interface resistance). The best match is found when the diffusivity factor is reduced to a 10 % of its original value, that is 0.10 × 𝛾𝑎𝑑𝑠,𝐿𝑀𝐽. Additionally, for the M4 configuration, different 𝛾 values were needed to account for the incremental effect of successive interfaces. The best fit was obtained with 0.10 × 𝛾𝑎𝑑𝑠,𝐿𝑀𝐽 for the first interface (to be consistent with the previous case) and 0.02 × 𝛾𝑎𝑑𝑠,𝐿𝑀𝐽. Table 5.3. Input parameters used for the different masonry modelling strategies (shaded cells show calibration parameters). Modelling strategy Case study Material/ Element Parameters 𝛾𝑎𝑑𝑠 𝑤𝑐𝑎𝑝 𝜓 𝑎 𝑚 𝐴𝑤.(1) 𝜇 𝑅𝐼𝐹 [–] [kg/m3] [–] [1/Pa] [–] [kg/(m2s0.5)] [–] [m/s] Detailed micro– M2 B 3.80 240 1.0070 – – 0.075 34.14 – LMJ 3.80 190 – 3.8E–6 0.352 0.080 15.00 – Int.1/2 – – – – – – – 2.0E+9 M4 B 3.80 240 1.0070 – – 0.075 34.14 – LMJ 3.80 190 – 3.8E–6 0.352 0.080 15.00 – Int.1/2 – – – – – – – 2.0E+9 Int.3/4 – – – – – – – 2.0E+10 Continuous micro– M2 B 3.80 240 1.0070 – – 0.075 34.14 – LMJ 0.40 190 – 3.8E–6 0.352 0.080 15.00 – M4 B 3.80 240 1.0070 – – 0.075 34.14 – LMJ1 0.40 190 – 3.8E–6 0.352 0.080 15.00 – LMJ2 0.08 190 – 3.8E–6 0.352 0.080 15.00 – Discrete micro– M2 Eq.B 3.80 234 1.0070 – – 0.076 31.84 – Int.1 – – – – – – – 4.0E+9 M4 Eq.B 3.80 232 1.0070 – – 0.076 31.21 – Int.1 – – – – – – – 4.0E+9 Int.2 – – – – – – – 4.0E+10 Macro– M2 Eq.B 3.04 234 1.0070 – – 0.076 31.84 – M4 Eq.B 3.04 232 1.0070 – – 0.076 31.21 – .(1) For 2–D models, B and Eq.B have orthotropic behaviour, 𝐴𝑤,𝑋= 0.104 kg/(m2s0.5) Numerical simulations I: Moisture transport 105 (a) (b) (c) (d) (e) (f) Figure 5.9. Capillary absorption simulated using different modelling strategies: (a) continuous micro–modelling, M2; (b) continuous micro–modelling M4; (c) discrete micro–modelling, M2; (d) discrete micro–modelling M4 (e) macro–modelling, M2; (f) macro–modelling, M4. 0100 200 300 400 500 600 700 0 10 20 30 40 50 60 0.10·  ads,LMJ Exp.M2 Exp.M2 Avg. Sim.M2 Water uptake mass [g] Time [sqrt(s)] Original  ads,LMJ Int.2 Int.1 0250 500 750 1000 1250 1500 0 15 30 45 60 75 90 0.02·  ads,LMJ,2 0.10·  ads,LMJ,1 0.10·  ads,LMJ Exp.M4 Exp.M4 Avg. Sim.M4 Water uptake mass [g] Time [sqrt(s)] Original  ads,LMJ Int.2 Int.1 Int.3 Int.4 0100 200 300 400 500 600 700 0 10 20 30 40 50 60 Exp.M2 Exp.M2 Avg. Sim.M2 Brick Sim.M2 Eq. Brick Water uptake mass [g] Time [sqrt(s)] Int.2 Int.1 Perfect contact RIF 4E+09 m/s 0250 500 750 1000 1250 1500 0 15 30 45 60 75 90 RIF,2 4E+10 m/s RIF,1 4E+09 m/s Exp.M4 Exp.M4 Avg. Sim.M4 Brick Sim.M4 Eq. Brick Water uptake mass [g] Time [sqrt(s)] Perfect contact RIF,1-2 4E+09 m/s Int.2 Int.1 Int.3 Int.4 0100 200 300 400 500 600 700 0 10 20 30 40 50 60 1.00·  ads Exp.M2 Exp.M2 Avg. Sim.M2 Brick Sim.M2 Eq. Brick Water uptake mass [g] Time [sqrt(s)] 0.80·  ads Int.2 Int.1 0250 500 750 1000 1250 1500 0 15 30 45 60 75 90 Exp.M4 Exp.M4 Avg. Sim.M4 Brick Sim.M4 Eq. Brick Water uptake mass [g] Time [sqrt(s)] 0.80·  ads 1.00·  ads Int.2 Int.1 Int.3 Int.4 Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 106 In terms of requirements, the same input material parameters demanded by the detailed micro–modelling approach are needed for the continuous micro–model. The latter does not consider hydraulic interfaces and therefore the calibration must be done through the diffusivity factor 𝛾. Since there are no interfaces, the complexity of the model is relatively low. In general, the continuous micro–modelling approach can provide good accuracy of results and it may be recommended in cases where interfaces need to be explicitly avoided. 5.4.2 Discrete micro–modelling The results for water absorption simulations using the discrete micro–modelling strategy (Figure 3.4c) are presented in Figure 5.9c and Figure 5.9d. The necessary parameters and updated material properties are presented in Table 5.3. Considering the discrete micro–modelling strategy, masonry was represented by the combination of two components: the bricks were modelled as a continuum with equivalent material properties and extended size, whereas interfaces were used to simulate the presence of mortar joints and brick–mortar interfaces. If the properties of both materials are known, the volume fraction of each material may be used to define equivalent properties. Otherwise, brick properties may be assumed without compromising the accuracy (see curves using brick versus equivalent brick properties in Figure 5.9c and Figure 5.9d). The main parameter used for calibration within the discrete micro–model strategy are the hydraulic resistances, 𝑅𝐼𝐹, imposed at the interfaces. It was found that a value 𝑅𝐼𝐹= 4.0E+9 m/s provided a good fit for the M2 configuration. Note that this value is double the resistance originally used for the detailed micro–model. This is expected since now a single interface must stand for the two interfaces of the detailed micro–model. On the other hand, two sets of values were used to match the experimental results of M4 configuration, namely 𝑅𝐼𝐹= 4.0E+9 m/s for the first interface (consistent with the M2 case) and 𝑅𝐼𝐹= 4.0E+10 m/s for the second interface. The higher value for the second interface is consistent with the hypothesis of an in–series interfacial phenomenon, that is the additive effect of successive interfaces. Once again, it is noted that the value imposed at the second interface is double the resistance used for the corresponding detailed micro–modelling case. In terms of requirements, the discrete micro–modelling approach has the advantage of needing fewer input material parameters than the other micro–modelling approaches. If the properties of brick and mortar are known, an equivalent material may be calculated by volume averaging as presented in Eq. (5.16). Otherwise, the model may produce considerably accurate results assuming only the properties of the brick (compare the curves produced using brick versus equivalent brick properties in Figure 5.9c Numerical simulations I: Moisture transport 107 and Figure 5.9d). This simplification is valid as long as the volume proportion of masonry units is higher than the corresponding volume of mortar joints, which is usually the case in brickwork masonry. Due to the extended geometry of the bricks, the geometric definition of the system may be somewhat more complex than the other strategies. On the other hand, the model is simplified by assuming only one material. Moreover, interfaces can capture localized phenomena such as the imperfect contact between adjacent layers. Overall, the discrete micro–modelling technique provides high accuracy and is a good compromise between simplicity and quality of results. Its use is encouraged when mortar properties are not known. 5.4.3 Macro–modelling Finally, water absorption simulations were performed considering a macro–modelling strategy (Figure 3.4d). The results are shown in Figure 5.9e and Figure 5.9f. Following the macro–modelling approach, masonry was idealized as a continuum with equivalent material properties. On this occasion, the homogenized medium must capture the overall behaviour of the system so that it provides an average trend. As in the discrete micro–modelling case, the volume fraction of each material may be used to define equivalent properties if both brick and mortar parameters are known. Otherwise, brick properties may be applied to the system and similar results can be obtained (see curves using brick versus equivalent brick properties in Figure 5.9e and Figure 5.9f). The diffusivity factor 𝛾 was used as a calibration parameter for curve–fitting. The curves obtained with the original factor 𝛾𝑎𝑑𝑠= 3.80 were able to capture the initial water uptake but largely overestimated the absorption rate as time evolved. This was expected since it is the usual response of a monolithic specimen (see absorption of single materials in Figure 5.2). For the M2 configuration, a reduction of the diffusivity parameter, e.g. 0.80 × 𝛾𝑎𝑑𝑠, was needed to produce a water uptake curve that could fit the overall experimental envelope. This type of correction provides an average approximation: the initial absorption rate is underestimated, whereas the absorption rate for prolonged times is overestimated. Note that the same reduction applied to M4 cases is not quite satisfactory since it deviates considerably from the experimental results. Macro–modelling approaches are commonly used for global structural analyses and their application to moisture transport is conveniently straightforward. The geometry is very simple and only one material is modelled. Moreover, the properties of brick and mortar may be used to define an equivalent medium or solely brick properties may be considered. Nonetheless, this strategy proves very little flexibility, and the simulation cannot capture the changes in moisture flux with evolving time or localized phenomena Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 114 effect) as well as moisture flux induced by temperature gradients (Soret effect) are considered negligible. The coupling between heat and moisture fields will be further discussed in the following sections. The effect of imperfect contact interfaces in multi–layered cases is considered for moisture transport by means of hydraulic resistances as presented in Chapter 5. For heat transfer, the impact of interfaces between adjacent materials is disregarded so that the temperature profiles are continuous between layers. 6.1.1 Governing equations The heat transfer model considers conductive heat transfer according to Fourier’s Law and accounts for the heat of vaporization: 𝜌𝐶𝜕𝑇 𝜕𝑡=∇·(𝜆 ∇𝑇)+𝐿𝑣 ∇ · (𝛿𝑎 𝜇 ∇(𝜑 𝑝𝑣,𝑠𝑎𝑡)) (6.1) where 𝜌𝐶 [J/(m3·K)] is the volumetric heat capacity at constant pressure, 𝑇 is the temperature [K], 𝑡 [s] is time, 𝜆 [W/(m·K)] is the thermal conductivity, 𝐿𝑣 [J/kg] is the latent heat of evaporation, 𝛿𝑎 [kg/(m·s·Pa)] is the water vapour permeability of still air, 𝜇 [–] is the water vapour resistance of the material, 𝜑 [–] is the relative humidity, and 𝑝𝑣,𝑠𝑎𝑡 [Pa] is the vapour saturation pressure. The second term on the right–hand side of Eq. (2.1) represents the latent heat of vaporization associated with liquid/gas phase change. On the other hand, the moisture transport equation presented in Chapter 5 is recalled here for clarity: 𝜕𝑤 𝜕𝜑𝜕𝜑 𝜕𝑡=∇·(𝜕𝑤 𝜕𝜑𝐷𝑤 ∇𝜑)+ ∇ · (𝛿𝑎 𝜇 ∇(𝜑 𝑝𝑣,𝑠𝑎𝑡)) (6.2) where 𝑤 [kg/m3] is the moisture content, and 𝐷𝑤 [m2/s] is the liquid water diffusivity. The remaining parameters are the same as presented in Eq. (2.1). The modified version of the exponential expression proposed by Künzel (1995) is used to describe the liquid water diffusivity: 𝐷𝑤=𝛾·(𝐴𝑤 𝑤𝑐𝑎𝑝)2·103·( 𝑤 𝑤𝑐𝑎𝑝 −1) (6.3) where 𝛾 [–] is a diffusivity factor depending on the material and transport process (wetting/drying), 𝐴𝑤 [kg/(m2·s0.5)] is the capillary absorption coefficient, and 𝑤𝑐𝑎𝑝 [kg/m3] is the capillary moisture content. Numerical simulations II: Hygro–thermo–mechanical coupling 115 6.1.2 Hygrothermal coupling The variety of relations and inter–dependencies between the heat and moisture fields are discussed in this section. First, the moisture dependency of heat transfer parameters is addressed. Then, the thermal effects on mass transport properties are discussed. Moisture dependency of thermal parameters To begin with, the heat capacity of a porous material is directly related to its moisture content. In particular, if the material is wet, the effect of moisture on heat capacity can be taken into account: 𝜌𝐶=𝜌𝑏𝑢𝑙𝑘𝐶𝑝+𝑤·𝐶𝑤 (6.4) where 𝜌𝐶 [J/(m3·K)] is the volumetric heat capacity of the material, 𝜌𝑏𝑢𝑙𝑘 [kg/m3] is the bulk density of the material, 𝐶𝑝 [J/(kg·K)] is the specific heat capacity of the dry material, 𝑤 [kg/m3] is the moisture content, and 𝐶𝑤 [J/(kg·K)] is the specific heat capacity of liquid water. The specific heat capacity of water has a weak thermal dependence for the range of temperatures of interest in this study. Therefore, it is taken as a constant, 𝐶𝑤= 4182 J/(kg·K), which corresponds to the specific heat capacity of water at 20 ℃. For a detailed list of saturated water properties, the reader is referred to Appendix 1 at the end of this document. Additionally, the moisture influence on the thermal conductivity of a wet porous material can be determined through the following expression: 𝜆=𝜆0 (1+𝑏·𝑤 𝜌𝑏𝑢𝑙𝑘 ⁄)=𝜆0 (1+𝑏·𝑤𝑔) (6.5) where 𝜆0 [W/(m·K)] is the thermal conductivity of the solid matrix, and 𝑏 [–] is a material–dependent thermal conductivity supplement. The supplement 𝑏 is commonly expressed as a percentage as well, so that it represents the fractional increase [in %] of the thermal conductivity per mass–% moisture content, i.e. moisture mass per unit mass of dry material × 100. Thermal conductivity has a minor dependence on temperature for normal environmental conditions and therefore its thermal dependency is disregarded. For detailed information about the thermal behaviour of porous building materials, the reader is referred to specialized literature, e.g. Cammerer (1995). For practical applications, the thermal conductivity of porous materials is often given as a linear relationship based on experimental values determined for different moisture content conditions. For instance, an expression of the form 𝜆=𝐴+𝐵·𝑤 has been extensively used in the literature to describe the thermal conductivity of brick (Hagentoft et al., 2003; Janssen et al., 2007; Defraeye et al., 2013). In the previous expression, the intercept 𝜆(𝑤= 0) stands for the thermal conductivity of the dry material, Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 116 𝜆0. In turn, the slope of the curve represents the increase in thermal conductivity per moisture content and according to Eq. (6.5) it must equal 𝜆0·𝑏/𝜌𝑏𝑢𝑙𝑘. Temperature dependency of hygric parameters The coefficient of thermal expansion of most building materials lies in the order of 1.0E–5 to 1.0E–6 ℃−1 for normal temperatures. Consequently, any volumetric change induced by temperature variations within the normal temperature range falls outside ordinary measurement capabilities (Feng & Janssen, 2016). Thus, it is generally accepted that open porosity, 𝜙𝑜 [–], and bulk density, 𝜌𝑏𝑢𝑙𝑘 [kg/m3], can be considered as constant for the range of temperatures of interest in this study (Salager et al., 2007). Moreover, once the volumetric variation of the porous solid matrix is disregarded, the influence of temperature on the saturation moisture content, 𝑤𝑠𝑎𝑡 [kg/m3], stems solely from the water density, which has a weak dependency on temperature (see Appendix 1), with less than a 1.20 % decline between 0 ℃ and 50 ℃. As a result, 𝑤𝑠𝑎𝑡 [kg/m3] can be considered as a constant independent of temperature. In the present study, an analogous logic is followed for the definition of the capillary moisture content, 𝑤𝑐𝑎𝑝 [kg/m3], which is considered a constant. However, it must be mentioned that a widely accepted theory to describe the relationship between 𝑤𝑐𝑎𝑝 and temperature is still lacking in the literature. Apart from the saturation state boundaries, 𝑤𝑐𝑎𝑝 and 𝑤𝑠𝑎𝑡, the equilibrium moisture content of unsaturated porous materials is dependent on the temperature. The thermal dependency of moisture storage implies a shift in the moisture storage function (sorption isotherm or water retention curve). This behaviour is expected since higher temperatures are associated with higher energy levels, which cause faster transport and release of water molecules (K. K. Hansen, 1986). For this reason, both adsorption and desorption curves corresponding to higher temperatures lie below the ‘colder’ isotherms (Pavlík et al., 2012). The thermal dependency of moisture storage can be expressed analytically through temperature–dependent fitting parameters. As an example, the original function proposed by Künzel (see Eq.(4.1)) can be rewritten as (Castellazzi et al., 2014): 𝑤(𝜑)=𝑤𝑐𝑎𝑝·𝜓(𝑇)−1 𝜓(𝑇)−𝜑·𝜑 (6.6) where 𝜓(𝑇) [–] is a fitting parameter dependent on temperature. It is noted that 𝜓 is inversely proportional to the temperature, i.e. 𝜓 decreases with increasing temperature. The correlation between 𝜓 and 𝑇 must be established from experimental data. However, the influence of temperature on moisture storage has not been sufficiently studied and experimental data are still scarce in the literature. Nonetheless, the available information suggests a negligible temperature dependency for barely Numerical simulations II: Hygro–thermo–mechanical coupling 117 hygroscopic materials, such as fired–clay brick (Feng & Janssen, 2016). For the sake of simplicity, the thermal effects on moisture storage are not considered in the simulations presented in this thesis. The influence of temperature on vapour diffusion is partly considered through the water vapour permeability of still air, 𝛿𝑎 [kg/(m·s·Pa)], which was introduced in Chapter 3 and is recalled here: 𝛿𝑎=2.31·10−5 𝑅𝑣·𝑇 𝑝0 𝑝(𝑇 273.15)1.81 (6.7) where 𝑝0= 101325 Pa is the standard atmospheric pressure, 𝑝 [Pa] is the ambient barometric pressure, 𝑅𝑣= 461.5 J/(kg·K) is the universal gas constant for water vapour, and 𝑇 [K] is the temperature. Additionally, the saturation vapour pressure, 𝑝𝑣,𝑠𝑎𝑡 [Pa], is a non–linear function of temperature, as it was defined in Chapter 3: 𝑝𝑣,𝑠𝑎𝑡=610.7[Pa]·107.5(𝑇−273.15 𝑇−35.85) (6.8) The impact of temperature on the liquid transport term has been studied by different authors, in particular its effect on the water absorption coefficient, 𝐴𝑤 [kg/(m2·s0.5)], or sorptivity, 𝑆=𝐴𝑤/𝜌𝑤 [m/s0.5], e.g. Gummerson et al. (1980), Guizzardi et al. (2016), Feng & Janssen (2016, 2017), Hanumanthu & Sarkar (2021). The reported results support the validity of the Lucas–Washburn law, which describes the water penetration depth 𝑥 [m] after a period of time 𝑡 [s] (Washburn, 1921): 𝑥=√ 𝜎 𝑟𝑐𝑜𝑠𝜃 2 𝜂 ·√𝑡 (6.9) where 𝜎 [N/m] is the surface tension of water, 𝑟 [m] is the equivalent hydraulic capillary radius, 𝜃 is the contact angle, and 𝜂 [Pa·s] is the viscosity of water. The previous expression states that liquid water flow is influenced by a series of factors. In order to understand how variations of temperature affect the overall liquid flow, it is convenient to study the impact of temperature on each of those factors separately. First, assuming that the porous structure undergoes negligible changes due to thermal (or hygric) expansion, it is reasonable to assume a constant 𝑟. Moreover, temperature has only a slight impact on the contact angle 𝜃 (see Appendix 1), so its influence can be disregarded. Therefore, the remaining temperature– dependent parameters are the surface tension 𝜎 and the viscosity 𝜂. Finally, by comparing the definition of the capillary absorption coefficient with the Lucas–Washburn equation, it is possible to establish a relation of proportionality as suggested by (Gummerson et al., 1980): 𝐴𝑤∝√𝜎 𝜂 ⁄ (6.10) Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 118 Considering the previous relation, Feng & Janssen (2016) fitted tabulated values for 𝜎 and 𝜂 (valid for temperatures between 0 ℃ and 50 ℃) and established a linear model to describe the relationship between 𝐴𝑤 and 𝑇: 𝐴𝑤=𝑘 √𝜎 𝜂 ⁄=𝑘 [0.095(𝑇−273.15)+6.566] (6.11) where 𝑘 [kg/m2.5] is a microstructure–dependent but temperature–independent factor. As an example, these authors studied the absorption behaviour of different building materials under several temperature conditions and determined 𝑘= 0.0059 for brick (𝐴𝑤= 0.052 at 22.5 ℃). For cement mortars, Hanumanthu & Sarkar (2021) found 𝑘= 0.0019–0.0023 (𝐴𝑤= 0.016–0.020 at 20 ℃), whereas for NHL mortar, the data presented by Karagiannis et al. (2016) were reinterpreted by Feng & Janssen (2017), who established 𝑘= 0.0125 (𝐴𝑤= 0.106 at 20 ℃). 6.1.3 Initial and boundary conditions The hygrothermal model is completed by introducing the initial and boundary conditions. The Dirichlet boundary condition yields: 𝑇=𝑇 (6.12) 𝜑=𝜑 (6.13) where 𝑇 [K] and 𝜑 [–] are the prescribed temperature and relative humidity at the boundary, respectively. Conversely, the Neumann boundary condition is defined as a flux derived from a temperature or vapour pressure difference, such as: 𝑞=ℎ𝑇(𝑇−𝑇𝑒𝑥𝑡) (6.14) 𝑔=ℎ𝑚(𝑝𝑣−𝑝𝑣,𝑒𝑥𝑡) (6.15) where 𝑞 [W/m2] is the convective heat flux, ℎ𝑇 [W/(m2·K)] the heat transfer coefficient, 𝑇𝑒𝑥𝑡 [K] is the temperature of the environment and 𝑇 [K] is the temperature at the boundary, 𝑔 [kg/(m2·s)] is the convective moisture flux, ℎ𝑚 [s/m] is the convective mass transfer coefficient, 𝑝𝑣,𝑒𝑥𝑡 [Pa] and 𝑝𝑣 [Pa] are the partial vapour pressures defined for the environment and at the boundary, respectively. In the present work, ℎ𝑇 accounts for the combined effects of convection and long–wave radiation exchanges with the environment. Thus, this lumped convective/radiative heat transfer coefficient is defined as: ℎ𝑇=ℎ𝑇,𝑐+ℎ𝑇,𝑟 (6.16) Numerical simulations II: Hygro–thermo–mechanical coupling 119 where ℎ𝑇,𝑐 [W/(m2·K)] is the convective heat transfer coefficient, and ℎ𝑇,𝑟 [W/(m2·K)] is the radiative heat transfer coefficient. The heat transfer coefficient is dependent on the local air flow conditions, the temperature, the geometry and the orientation of the studied element, which makes an accurate estimation significantly complex. For building applications, a simplified approach is usually adopted, and the heat transfer coefficient is assumed to be constant (Künzel, 1995). 6.2 HYGROTHERMAL SIMULATIONS OF A BRICK MASONRY WALL In this section, the model described in the previous paragraphs is used to simulate the hygrothermal behaviour of a brick masonry wall. First, the main features of the numerical model are described: geometry, material properties and boundary conditions. Then, two sets of analyses are performed, namely steady–state and time–dependent or transient analyses. Finally, the results of the simulations are presented in terms of temperature and relative humidity profiles, and the obtained trends are discussed. Like the moisture transport simulations presented in Chapter 5, the analyses described in the following paragraphs are based on the finite element method and were performed using the software COMSOL Multiphysics (COMSOL, 2021). 6.2.1 Description of the numerical model A common type of load–bearing masonry wall was selected as a case study to perform the hygrothermal simulations. In particular, the chosen wall is made up of bricks with dimensions 205 mm × 95 mm × 50 mm, and mortar joints 15 mm thick. The resulting structure is a 315 mm thick (three–wythe brick) masonry wall with a total height of 2650 mm. The geometrical configuration of the wall is depicted in Figure 5.2. Finally, the described geometry was used to prepare a two–dimensional (2–D) model. The material properties used for the hygrothermal simulations are presented in Table 2.2. Two types of mortar were considered for the analyses, namely natural hydraulic lime mortar (LMJ) and cement mortar (CM). Moreover, the brick–mortar interfaces were assumed to have perfect contact for heat transfer, whereas an imperfect contact was considered for moisture transport. Thus, a hydraulic resistance 𝑅𝐼𝐹= 2E+9 m/s was assigned to the brick–mortar interfaces. The wall is assumed located in Guimarães (Portugal), which has an annual average temperature 𝜗= 13.5 ℃; and an annual average relative humidity RH= 74 % ( Climate and Average Weather Year– Round in Guimarães, Portugal , 2022). Furthermore, the wall is considered to have initial conditions equal to these yearly average values, thus 𝜗0= 13.5 ℃ and 𝜑0= 0.74. One face of the wall is exposed to the exterior, whereas the opposite face is in contact with a conditioned interior space with constant temperature and relative humidity, respectively 20 ℃ and 50 % RH. It must be noted that the presented Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 120 (a) (b) (c) Figure 6.1. Schematic geometrical configuration of the brick masonry wall: (a) transversal cross–section; (b) vertical cross– section; (c) bond arrangement and brick blocks geometry. Dimensions in mm. configuration is rather unusual for a building envelope in contact with a modern habitable interior space since it lacks any type of dedicated moisture barrier or thermal insulation. Nonetheless, this configuration is expedient for the analyses at hand, in which thermal and moisture gradients are of interest. In the case of a masonry wall with good insulation on the interior side, adiabatic boundary conditions could be assumed for the internal face. Table 6.1. Summary of material properties used for the hygrothermal simulations. Material Property Symbol Value Units Source B Bulk density 𝜌𝑏𝑢𝑙𝑘 1900 kg/m3 Experimental Open porosity 𝜙𝑜 0.280 – Experimental Capillary moisture content 𝑤𝑐𝑎𝑝 240 kg/m3 Experimental Fitting parameter for sorption isotherm 𝜓 1.0070 – Experimental Water absorption coefficient (∥ bed face) 𝐴𝑤,∥ 0.104 kg/(m2·s0.5) Experimental Water absorption coefficient (⊥ bed face) 𝐴𝑤,⊥ 0.075 kg/(m2·s0.5) Experimental Water vapour resistance factor 𝜇 34.14 – Experimental Specific heat capacity 𝐶𝑝 825 J/(kg·K) Kočí et al. (2018) Thermal conductivity 𝜆0 0.59 W/(m·K) Kočí et al. (2018) Thermal conductivity supplement 𝑏 12.74 – Kočí et al. (2018) Exterior Interior Numerical simulations II: Hygro–thermo–mechanical coupling 121 Table 6.1 (Continued). Summary of material properties used for the hygrothermal simulations. Material Property Symbol Value Units Source LMJ Bulk density 𝜌𝑏𝑢𝑙𝑘 2060 kg/m3 Experimental Open porosity 𝜙𝑜 0.230 – Experimental Capillary moisture content 𝑤𝑐𝑎𝑝 190 kg/m3 Experimental Fitting parameter for adsorption isotherm 𝑎𝑎𝑑𝑠 3.80E–6 1/Pa Experimental 𝑚𝑎𝑑𝑠 0.352 – Experimental Fitting parameter for desorption isotherm 𝑎𝑑𝑒𝑠 1.19E–5 1/Pa Experimental 𝑚𝑑𝑒𝑠 0.235 – Experimental Water absorption coefficient (isotropic) 𝐴𝑤,𝐼𝑆𝑂 0.080 kg/(m2·s0.5) Experimental Water vapour resistance factor 𝜇 15.00 – Prangnell (1971) Specific heat capacity 𝐶𝑝 840 J/(kg·K) Kumaran (1996) Thermal conductivity 𝜆0 0.85 W/(m·K) Kumaran (1996) Thermal conductivity supplement 𝑏 9.53 – Kumaran (1996) CM Bulk density 𝜌𝑏𝑢𝑙𝑘 2000 kg/m3 Experimental Open porosity 𝜙𝑜 0.210 – Experimental Capillary moisture content 𝑤𝑐𝑎𝑝 180 kg/m3 Experimental Fitting parameter for adsorption isotherm 𝑎𝑎𝑑𝑠 1.43E–6 1/Pa Experimental 𝑚𝑎𝑑𝑠 0.285 – Experimental Fitting parameter for desorption isotherm 𝑎𝑑𝑒𝑠 2.51E–5 1/Pa Experimental 𝑚𝑑𝑒𝑠 0.213 – Experimental Water absorption coefficient (isotropic) 𝐴𝑤,𝐼𝑆𝑂 0.060 kg/(m2·s0.5) Experimental Water vapour resistance factor 𝜇 25.00 – Prangnell (1971) Specific heat capacity 𝐶𝑝 932 J/(kg·K) Kumaran (1996) Thermal conductivity 𝜆0 1.72 W/(m·K) Kumaran (1996) Thermal conductivity supplement 𝑏 9.29 – Kumaran (1996) The hygrothermal simulations were performed through both steady–state (SS) and time–dependent (TD) analyses. The SS analysis was used to evaluate the final hygrothermal equilibrium attained by the wall considering constant environmental conditions as explained in the previous paragraph. Moreover, two sets of time–dependent analyses were defined. In particular, TD1 was performed to evaluate the evolution of temperature and relative humidity profiles across the wall from the initial state to hygrothermal equilibrium with the environment. Therefore, the final stage of TD1 is analogous to the results obtained with the stationary analysis SS. Additionally, the transient study TD2 considered the wall already in Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 122 equilibrium with the environment (the initial conditions were taken from the results of SS or the final stage of TD1) and then imposed variable external conditions. In particular, the external conditions for TD2 were assumed to vary with time following a sinusoidal law such as: 𝜗(𝑡)=13.5+10·cos(2𝜋𝑡−𝜋/2) (6.17) 𝜑(𝑡)=0.74+0.12·cos(2𝜋𝑡+𝜋/2) (6.18) for the external temperature and relative humidity, respectively, with 𝑡 expressed in days. The proposed thermal variation corresponds to the average maximum temperature excursion that can be experienced in a single day during mid–season periods, i.e. ∆𝜗≈ 20 ℃. Additionally, average values of the seasonal relative humidity were selected to determine the range of daily moisture variation and the corresponding sinusoidal function was set to follow an opposite trend to the one defined for temperature. The initial and boundary conditions used for the hygrothermal simulations of the masonry wall are summarised in Table 6.2. Neumann boundary conditions were imposed on the external and internal faces of the wall. Therefore, heat and moisture fluxes were defined with commonly accepted values for the convective transfer coefficients. In particular, heat transfer coefficients ℎ𝑇,𝑒𝑥𝑡= 25 W/(m2·K) and ℎ𝑇,𝑖𝑛𝑡= 8 W/(m2·K) were imposed at the external and internal faces, respectively (Hagentoft et al., 2004). It is noted that the heat transfer coefficient selected for the external surface already takes into account the radiative contribution. On the other hand, convective mass transfer coefficients ℎ𝑚,𝑒𝑥𝑡= 2.0E–7 s/m and ℎ𝑚,𝑖𝑛𝑡= 3.0E–8 s/m were considered for the external and the internal faces of the wall, respectively (Hagentoft et al., 2004). Table 6.2. Summary of the initial and boundary conditions used for the hygrothermal simulations. Analysis Initial conditions Exterior boundary conditions Interior boundary conditions 𝜗0 [℃] 𝜑𝑜 [–] 𝜗𝑒𝑥𝑡 [℃] ℎ𝑇,𝑒𝑥𝑡 [W/(m2 K)] 𝜑𝑒𝑥𝑡 [–] ℎ𝑚,𝑒𝑥𝑡 [s/m] 𝜗𝑖𝑛𝑡 [℃] ℎ𝑇,𝑖𝑛𝑡 [W/(m2 K)] 𝜑𝑖𝑛𝑡 [–] ℎ𝑚,𝑖𝑛𝑡 [s/m] SS 13.5 0.74 13.5 25 0.74 2.0E–7 20.0 8 0.50 3.0E–8 TD1 13.5 0.74 13.5 25 0.74 2.0E–7 20.0 8 0.50 3.0E–8 TD2 SS results.(1) SS results.(1) Eq. (6.17) 25 Eq. (6.18) 2.0E–7 20.0 8 0.50 3.0E–8 .(1) Analogous to the final stage of TD1 Two–dimensional quadrilateral elements with quadratic shape functions were used for the discretization of the models. After preliminary sensitivity analyses to determine the optimal mesh size, the average element size was set to 5 mm. The primary variables, namely temperature and relative humidity, were Numerical simulations II: Hygro–thermo–mechanical coupling 123 interpolated based on standard Lagrangian shape functions. Moreover, the backward finite difference method was used for automatic time discretization during the transient analyses. Finally, an iterative procedure based on the Newton–Raphson method was applied to solve the non–linear differential equations. 6.2.2 Hygrothermal behaviour of a brick masonry wall The results of the hygrothermal analyses presented in the previous section are examined in this sub– section. First, the steady–state case is discussed, with special emphasis on the temperature and moisture content distributions obtained for the average environmental conditions. Subsequently, the time– dependent studies are analysed, and the obtained results are assessed in terms of temperature and moisture content evolution across the wall and for different points of the external and internal surfaces. For each group of analyses, a distinction is made between the brick masonry wall with lime mortar (LMJ) and the one with cement mortar (CM). Steady–state analysis (SS) The temperature and relative humidity profiles obtained for the steady–state analyses are shown in Figure 5.3 and Figure 6.3, respectively. The results show only minor differences between the two studied cases, namely the wall with lime mortar (LMJ) and the wall with cement mortar (CM). This is expected considering that the stationary analysis focuses on the final equilibrium state, which is mostly dependent on the external boundary conditions. Moreover, any difference observed in the response of the wall stems from the different behaviour of the selected type of mortar, which ultimately accounts for less than 26 % of the total cross–sectional area. It is noted that the temperature and relative humidity values obtained at the external and internal surfaces do not reach the values defined for the boundary conditions, namely 13.5 ℃ and 74 % RH for the exterior, and 20 ℃ and 50 % RH for the interior. This is due to the kind of boundary conditions imposed on the model, i.e. convective flux or Neumann boundary conditions. Therefore, the convective heat transfer and the convective mass transfer coefficients induce, respectively, a temperature and a relative humidity exchange between the environment and the surface of the wall. Regarding the thermal field (see Figure 5.3), the temperature profiles for both cases show an overall linear distribution across the thickness of the wall. In a multi–layered material, the transition between adjacent layers with dissimilar thermal conductivity translates into a change of slope in the temperature profile before and after the interface. For the LMJ case, the presence of mortar joints is barely noticeable since the thermal conductivity properties of bricks and lime mortar are quite similar. On the contrary, the Hygro–thermo–mechanical analysis of masonry: Experimental characterization and numerical simulations 226 Belleudy, C., Woloszyn, M., Chhay, M., & Cosnier, M. (2016). 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